Schedule

Minisymposium Organisers

David Moxey, University of Exeter
Joaquim Peiró, Imperial College London

Abstract

Despite the increasing popularity of high-order methods in recent years, a significant barrier in the adoption of these methods is overcoming the challenge of generating curvilinear meshes that align with the underlying geometry. This is an important problem to overcome, since a valid mesh that accurately represents the underlying geometry is a prerequisite to running simulations and exploiting the potential of high-order methods for challenging industrial and academic problems.

The generation of curved meshes is typically done in an a posteriori fashion, by first generating a coarse linear mesh and then deforming elements connected to the boundary in order to align elements to the geometry. Invariably, however, this combination results in the generation of elements that self-intersect and are therefore unsuitable for simulations. Additionally, the projection used to impose the curvature on the element may yield inaccuracies in representing the 'true' CAD surface that defines the geometry. Recent work to solve these issues has focused on developing techniques based on solid body analogies or energy minimisations to deform the interior of the grid to accommodate boundary curvature and improve the accuracy of the surface representation. However, many challenges remain in, e.g. increasing robustness, generation of boundary layer grids and improving computational efficiency, amongst other issues.

The purpose of this minisymposium is to provide a platform for researchers in the field of high-order to present the latest developments in this area and progress in addressing these issues. It will offer the opportunity discuss the latest methods related to the generation of curvilinear meshes, techniques assessing the quality and validity of generated grids, assess the application of these mesh generation techniques to real-world problems in computational engineering and fluid dynamics, and provide a means through which researchers can collaborate on new developments in the field.

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Authors (presenter in bold)

Jean-Francois Remacle, Universite catholique de Louvain
Amaury Johnen, Universite catholique de Louvain
Ruili Zhang, Universite catholique de Louvain

Abstract

Assume a smooth function $f(x,y)$ defined in the unit square $(x,y) \in [0,1] \times [0,1]$. The simple question we aim to address here is the following: how can we build a quadratic curvilinear mesh of this unit square that contains $N$ triangles and that allows to minimize the interpolation error $$e = \|f - Pf\l.$$ Here, $Pf$ is the finite element interpolation (Clement Interpolant) of $f$ on the curvilinear mesh and $\| \cdot \|$ is either the $L^2$ or the $H^1$ norm.

We first compute a metric field $M(x,y)$ that allows to define local mesh lengths at any point of the domain. This field depends on the derivatives of $f$ up to order $3$. Then, corners of the triangulation are distributed in a frontal fashion. Then, points are connected with a standard anisotropic mesh and edges are curved in order to follow geodesics of the Riemannian space defined by $M$. This procedure allows to generate meshes that are naturally curvilinear and that adapt optimally to $f$.

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Authors (presenter in bold)

Amaury Johnen, Université catholique de Louvain, Barcelona Supercomputing Center
Xevi Roca, Barcelona Supercomputing Center
Thomas Toulorge, MINES ParisTech
Jean-Francois Remacle, Université catholique de Louvain

Abstract

We propose a new framework for curving any structured boundary-layer (BL) mesh (that we define as a BL mesh constructed by extruding the boundary and splitting the created elements if simplicial elements are requested). We curve each stack of elements separately and we allow the surface between the BL and the external mesh to be curved which is useful to control the thickness of the first layers. We make the choice to fix the corner of BL elements and we impose the mapping of the elements to be linear in the direction of extrusion. We first compute a curving of the exterior surface such that the distance to the boundary is constant in the least-square sense. This curving induce a deformation that can break the validity or significantly decrease the quality of the external mesh. When it is the case, the deformation is reduced such that to recover the validity and quality of the external mesh. The interior of the BL is then curved by using a linear transfinite interpolation or by solving an elliptic problem. In 3D, interfaces between stacks are curved before curving the stacks which ensures a straightforward parallelization of the whole process. In order to better control the thickness of the first layers, we compute the curving of the first layer surface in the same way we compute the curvature of the exterior surface and use this information for doing a Hermite transfinite interpolation. In pathologic cases, this can produce invalid elements in the stack in which case we compute a combination of the linear and Hermite transfinite interpolation. This method is extremely fast, produce high-quality curved BL mesh and is able to handle complex BL mesh. While not fully robust, an a posteriori optimization process allows to recover invalid stacks.

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Authors (presenter in bold)

Abel Gargallo-Peiró, Barcelona Supercomputing Center
Guillaume Houzeaux, Barcelona Supercomputing Center
Xevi Roca, Barcelona Supercomputing Center

Abstract

A parallel distributed approach to refine a mesh while preserving the curvature of a target geometry is presented. Our approach starts by generating a coarse linear mesh of the computational domain. Second, the former coarse mesh is curved to match the curvature of the target geometry. Then, the curved mesh is partitioned and the subdomain meshes are sent to the slaves. Finally, the curved elements are uniformly subdivided in parallel targeting the geometry approximated by the curved mesh. The result is a distributed finer linear or high-order mesh featuring improved geometric accuracy. The key ingredient of our implementation is to approximate the target geometry as a linear mesh equipped with an elemental field corresponding to an element-wise polynomial geometry representation. Thus, the distribution of the curved geometry is equivalent to partitioning the linear mesh and sending the subdomain meshes and the elemental fields to the slaves. The main application of the obtained finer mesh is to compute in parallel steady state flow solutions. The qualitative results show that steady state flow solutions our parallel subdivision approach mitigates the artificial artifacts that might appear with standard straight-sided subdivision methods. We also check the parallel performance of the implementation by performing a weak scalability test.

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Authors (presenter in bold)

David Moxey, University of Exeter
Julian Marcon, Imperial College London
Jan Eichstaedt, Imperial College London
Joaquim Peiró, Imperial College London

Abstract

Although high-order methods are continuing to make inroads into simulation practice, their use in industrial applications is predicated on the ability to generate valid high-order meshes. Commonly, such problems involve complex geometries, which pose several challenges for high-order mesh generation: the ability to generate sufficiently coarse and high quality linear meshes; converting these meshes to valid high-order meshes in an a posteriori manner; and the generation of boundary layers for applications involving fluid dynamics flows. In this presentation, we will outline the latest efforts in solving these problems in the high-order generator NekMesh, including GPU implementations for the efficient optimisation and correction of invalid meshes.

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Authors (presenter in bold)

Bernard Shizgal, University of British Columbia

Abstract

The interpolation of particular functions with a uniform grid on the interval [-1,1] is well known to diverge at the interval boundaries. Large oscillations occur at the interval boundaries with an increase in the order of the interpolation. This behavior is known as the Runge phenomenon and has been analyzed extensively [1]. By contrast to the uniform grid, the interpolation with the Chebychev quadrature points based on the Chebyshev polynomials is convergent. There have been numerous analyses of the convergence behaviour with interpolants based on radial basis functions [2], Hermite functions [3], Gaussian basis functions [4], Fourier continuation methods [5], cubic splines [6] and other methods. The present paper reports a study of the convergence of the interpolations with a quadrature grid defined by nonclassical polynomials orthogonal with respect to the weight function w(x) = 1/(1+ax^2)^(0.5), which for a = -1 defines the Chebyshev quadratures. The rate of convergence of the interpolation based on this class of weight functions is studied versus the parameter a in the weight function. The results are interpreted in terms of classical potential theory.

[1] L. N. Trefethen, Approximation Theory and Approximation Practice, (SIAM, 2013). [2] G. Fornberg, G. Wright, E. Larsson, Some observations regarding interpolants in the limit of flat radial basis functions, Comput. Math. Appl. 47, 37 (2004). [3] J. P. Boyd, Convergence and error theorems for Hermite function pseudo-RBFs: Interpolation on a finite interval by Gaussian-localized polynomials, Appl. Num. Math. 87, 125 (2015). [4] J. P. Boyd, Six strategies for defeating the Runge phenomenon in Gaussian radial basis functions on a finite interval, Comm. Comput. Phys. 60, 3108 (2010). [5] O. Bruno, Y. Han, M. M. Pohlman, Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis, J. Comput. Phys. 227, 1094 (2007). [6] Chen et al, Solving the Problem of Runge Phenomenon by Pseudoinverse Cubic Spline, IEEE 17th Int. Conf. Comput. Sci. Eng. 1226 (2014).

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Authors (presenter in bold)

Martin Almquist, Stanford University
Siyang Wang, Chalmers University of Technology, University of Gothenburg
Jonatan Werpers, Uppsala University

Abstract

Adaptive mesh refinement is essential for efficiency whenever high resolution in a localized area is desired. For wave-dominated phenomena, high-order finite difference (FD) methods are often computationally efficient, but not always robust. By combining summation by parts (SBP) operators with simultaneous approximation terms (SATs), the SBP-SAT methodology leads to energy stable and conservative high-order FD methods on multi-block and curvilinear grids.

Mattsson and Carpenter extended the SBP-SAT framework to locally refined grids by constructing SBP preserving interpolation operators for non-conforming grid interfaces. They proved energy stability for hyperbolic conservation laws and parabolic equations. Later, the technique was extended to the Schrödinger equation and the second order wave equation. For equations involving second derivatives, it was observed that the SATs at the non-conforming interfaces degrade the largest local truncation error, and hence the global convergence, by one order. Lundquist and Nordström showed that for first order equations, the larger local error decreases the global convergence by half an order. The obvious remedy would be to increase the order of the interpolation operators, but Lundquist and Nordström proved that this is impossible because the order of the interpolation operators is bounded from above by the order of the quadrature rule associated with every SBP operator. Friedrich et al. then constructed interpolation operators with order-preserving accuracy by developing novel degree preserving first derivative SBP operators, which are based on extra-high order quadrature rules. Unlike traditional SBP FD operators, the boundary closures of the degree-preserving operators depend on the number of grid points.

In this work we circumvent the order reduction without increasing the quadrature order. Instead, we modify the SATs at the non-conforming interface. For equations involving second derivatives, the key to improving the local truncation error is to introduce two pairs of interpolation operators. For first order hyperbolic equations, the key is to use a completely upwinded interface coupling. The new methods apply to any diagonal-norm SBP operator and retain the stability, dual-consistency, and conservation properties of SBP-SAT schemes for conforming interfaces. Numerical experiments with the Schrödinger, heat, and advection equations demonstrate the improved accuracy of the new method.

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Authors (presenter in bold)

Jonatan Werpers, Uppsala University

Abstract

Finite difference methods are known to be simple and efficient for solving partial differential equations on simple domains. For somewhat complicated domains the summation by parts framework together with weakly imposed boundary and interface conditions through the SAT technique provide tools for deriving provably stable high order schemes for a wide variety of equations, problems and boundary conditions. When the geometries get complicated enough it is often necessary to use some other method of discretisation. However, for large parts of the domain, a finite difference method might still be competitive.

In these cases, when the problem at hand demands adaptivity in mesh size or hybrid methods the traditional ways of using finite difference methods often come with certain costs. To decrease mesh size, or to change the method of discretisation in part of the domain, for stability reasons, one often has to introduce block interfaces in large parts of the domain even away from the particular area. Besides complicating the structure of the grid these interfaces also degrade accuracy in order to provide numerical stability of the scheme.

In this talk we will present summation by parts boundary closures around reentrant corners for a few different high order finite difference schemes. In combination with traditional summation by parts finite difference boundary closures for edges and regular interior corners this allows finite difference discretisations of non-rectangular domains without the introduction of block interfaces. Avoiding extra interfaces preserves the simplicity and efficiency of the method in the bulk of the domain while enabling provably stable couplings to other methods or finer grids. One example could be a large rectangular domain with a small square removed from its interior which is either not part of the problem domain or is discretised in a different way.

We will show that the new boundary closures yield provably stable interfaces in a similar way to traditional summation by parts operators and we will show that the schemes give higher order accuracy. We will also provide examples on how to do mesh refinement and hybrid schemes.

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Authors (presenter in bold)

Ylva Rydin, Uppsala University

Abstract

The time-dependent Schrödinger equation on deforming domains gives rise to interesting solutions that are caused by quantum mechanical phenomena. An example of such phenomenon is Berry phases. In this study, Berry phases on deforming quantum billiards are explored numerically with provably stable high-order finite difference methods to investigate how different physical parameters such as magnetic fields and non-linearities affect the system. To model the deforming quantum billiard, the Schrödinger equation is solved with Dirichlet boundary conditions on domains with moving boundaries.

Berry phases are related to degenerate eigenstates of two or more energy levels in quantum systems. In quantum billiards where the energy surface forms a double-cone connection in the parameter space of the system, the degeneracy is referred to as a ’diabolical point’. If a quantum system is deformed adiabatically encircling a diabolical point, a geometrical phase shifts of the wave function occur which is called a Berry phase. In this problem both the rapid high frequent wave functions and the slow adiabatic deformation has to be covered for an accurate model. Hence, two time scales are present in the simulations. This makes the problem numerically difficult and efficient methods are required.

It is well known that high order methods capture wave dominated phenomena efficiently since they allow a considerable reduction in the degrees of freedom for a given error tolerance. In particular, high-order finite difference methods are ideally suited for problems of this type. The main difficulty using finite differences in this setting is to achieve a stable boundary treatment at the moving boundaries.

In this talk, a high order provably stable finite difference approximation of the linear Schrödinger equation on deforming domains will be presented. Emphasis will be given on accurate treatment of domain deformations by a time dependent coordinate transform. The transformed system is discretised with high-order summation-by-parts (SBP) finite difference operatros. To achieve energy-stability the boundary conditions are imposed using a penalty technique. The discretisation leads to highly robust and accurate approximations, corroborated through numerical computations. The method is also verified by comparison to physical experiments.

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Authors (presenter in bold)

Jan Jaskowiec, Cracow University of Technolgy

Abstract

The hp-type mesh adaptation in the discontinuous Galerkin (DG) method is presented. The mesh refinement followed by an error estimation using DG methods is in the mainstream of current scientific research [1]. The very high order DG method is used to describe approximation orders in finite elements that exceed 20, and may reach up to 100. To achieve such a high order, the DG method with finite difference rules must be applied (DGFD) with Chebyshev polynomials as the basis functions [2,3]. For the mesh refinement, an a posteriori error estimation needs to be applied. In this paper, two techniques for error estimations are proposed. In the first technique, the additional boundary value problem must be solved and performed with minimal computational costs. In the second technique, the error is estimated locally for every element in the mesh. Both of these methods are quite simple and effective as they utilize some unique properties of the DGFD. The mesh in the DGFD method may be non-conforming in the hp-sense. This means that two adjacent elements may differ in size, as well as in their approximation orders. For example, a low order element can be placed next to a high order element, no matter their sizes. This exemplifies how the hp-type mesh refinement is very simple and flexible, and the mesh can be easily adjusted to the estimated error distribution. This paper is illustrated with a set of two-dimensional benchmark examples for Poisson’s as well as elasticity problems. In the examples, the hp mesh adaptations are presented using the two proposed techniques for error estimations. The fast convergence of the approximate solutions are achieved.

[1] Baccouch, M., 2017. A posteriori error estimates and adaptivity for the discontinuous Galerkin solutions of nonlinear second-order initial-value problems. Applied Numerical Mathematics 121, 18-37. [2] Jaśkowiec, J., 2017. Very high-order discontinuous Galerkin method in elliptic problems. Computational Mechanics, In press. [3] Jaśkowiec, J., 2017. Application of discontinuous Galerkin method to mechanical 2D problem with arbitrary polygonal and high-order finite elements. Computer Methods in Applied Mechanics and Engineering 323, 389-415.

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Authors (presenter in bold)

Yoshiaki Abe, Imperial College London
Peter E. Vincent, Imperial College London

Abstract

Many aeronautical problems, such as landing gear aeroacoustics and design optimisation of jet engine turbines, are characterized by highly unsteady and turbulent flow phenomena. It has been shown that high-order unstructured schemes, e.g., Discontinuos Galerkin (DG) and Flux Reconstruction (FR) methods, have the potential to accurately simulate these flows. However, to date, grids for large-eddy simulation (LES) or implicit LES (ILES) with high-order DG/FR schemes are typically constructed based on empirical criteria gained from “expert” experience, and often require many iterations.

This study aims to construct a framework to identify, a priori, grid properties that achieve accurate prediction of turbulent boundary layers when performing ILES with high-order FR schemes. Wall-normal cell distribution is defined by two parameters, \alpha and \beta, where \alpha determines the minimum wall-normal cell size adjacent to the wall, and \beta determines the expansion ratio of cell sizes in the wall-normal direction. An extensive parametric study is carried out in \alpha\beta-space on a turbulent plane Couette flow (Re_tau = 171) for various polynomial orders (p) and total degrees of freedom (DoF). Three cost functions are defined (log-layer mismatch, error in Re_\tau, and allowable time-step size). A Kriging surrogate model of each cost function is then constructed in \alpha\beta-space, which provides a continuous map of computational accuracy and cost, with respect to wall-normal cell distributions for high-order FR schemes. Finally, insights from the surrogate model are used to identify optimal meshing strategies for ILES of turbulent boundary layers with high-order FR schemes.

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Authors (presenter in bold)

Julian Marcon, Imperial College London
David Moxey, University of Exeter
Spencer J. Sherwin, Imperial College London
Joaquim Peiró, Imperial College London

Abstract

We propose a new approach to the simulation of flows with shocks based on dual r- and p-adaptation techniques. Previous approaches have involved a combination of h-adaptation for shock capturing and adaptive p-refinement for the smooth field regions. h-adaptation is usually performed through a range of mesh operations, such as element splitting and merging, edge swapping and collapsing and point insertion, allowing to obtain metric-aligned elements in the thickness of shocks. Adaptive p-refinement on the other hand is used for smooth field regions were exponential error decay rates can be achieved.

In this work, we substitute r- for h-adaptation, shifting into a moving mesh framework with constant number of degrees of freedom and preserved connectivities. This approach allows us to better target High-Performance Computing (HPC) systems where users want to harness all available computational resources and where node-to-node communication is a major bottleneck. This work relies on the Nektar++ platform where the Nektar++ solvers are used for adaptive p-refinement and NekMesh, its suite of high-order mesh generation tools, is used for r-adaptation. The latter relies on a variational framework recently developed to allow for the deformation of high-order meshes based on metric tensors.

To test this new approach, we analyse two well-known configurations. The first one is a low angle of attack NACA0012 aerofoil in transonic regime where a strong shock appears on the upper surface and a weak shock on the lower one. The second configuration corresponds to a bluff body in supersonic regime generating a large bow shock upstream of the body. In both cases we are able to achieve anisotropic deformation of the mesh in the vicinity of the shock in order to better capture it. We then apply adaptive p-refinement to the rest of the domain with most refinement observed in the regions upstream and downstream of the shock where the flow field is smooth but larger gradients appear. With this dual approach we are able to achieve increased accuracy in both smooth and discontinuous field regions without ever modifying mesh connectivity.

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Authors (presenter in bold)

Andrés Mauricio Rueda-Ramírez, Universidad Politécnica de Madrid
Gonzalo Rubio, Universidad Politécnica de Madrid
Esteban Ferrer, Universidad Politécnica de Madrid
Eusebio Valero, Universidad Politécnica de Madrid

Abstract

High-order discontinuous Galerkin methods are gaining increasing popularity in Computational Fluid Dynamics because of their spectral accuracy and flexibility. Nevertheless, their main drawback is that increasing the order of approximation does not only improve the accuracy exponentially, but it also increases the computational cost. Several techniques have been introduced to reduce this cost, such as local mesh adaptation strategies, which reduce the number of degrees of freedom without loss of accuracy; multigrid solvers, which accelerate time marching computations for a fixed number of degrees of freedom; among many others.

In this work, we present an anisotropic p-adaptation multigrid algorithm for steady-state problems that uses a multigrid scheme both as a solver and as an anisotropic error estimator. To achieve this, we develop a new anisotropic truncation error estimator based on the tau-estimation method, that can be evaluated inside the multigrid cycle with a negligible extra cost. We show that our new error estimator is cheaper to evaluate than previous versions of the tau-estimation procedure and that it obtains more accurate extrapolations of the truncation error for high-order representations. In our technique, a non-converged solution in a reference mesh is used to estimate the truncation error with the multigrid scheme for different combinations of polynomial orders inside every element, and the mesh is adapted accordingly to target a desired truncation error threshold. Furthermore, we introduce a multi-stage p-adaptation procedure that takes advantage of a Full Multigrid cycling strategy and performs multiple p-adaptation steps at increasing polynomial orders. This strategy reduces the computational time when very low truncation errors are required.

The new error estimation procedure is validated using the method of manufactured solutions, and the accuracy and computational cost of the proposed p-anisotropic adaptation algorithm is tested in 2D and 3D boundary layer simulations: a flat plate with a highly stretched mesh (Re=6000, M=0.2) and the flow around a sphere (Re=200, M=0.2). A speed-up of the order of 1000 is achieved for a fixed accuracy in the proposed numerical examples with respect to a non-locally adapted and explicitly time marched simulation.

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Authors (presenter in bold)

Vikram Singh, Technion - Israel Institute of Technology
Steven Frankel, Technion - Israel Institute of Technology

Abstract

High-order methods have become increasingly useful and popular in recent years. Flux reconstruction (FR) methods represent an arbitrarily high-order family of methods that employ unstructured body-fitted grids to enable simulations over and through complex geometries. Various studies have demonstrated that the inherent numerical dissipation associated with these schemes make flux reconstruction eligible for so-called implicit Large Eddy Simulation (ILES) . This obviates the need for a subgrid-scale turbulence model. FR schemes, similiar to closely related schemes like Discontinuous Galerkin (DG) and spectral-element methods, allow p-refinement beyond the typical h-refinement. This offers the possibility of adapting the order of accuracy within an element without the complexity of non-conformal cells. On the other hand, when using explicit time-stepping, p-refinement introduces time step restrictions due to its dependency on p. In the present study, we investigate the use of static and dynamics p-refinement for the Taylor Green Vortex (TGV) case, which undergoes transition to turbulence in time and the classic fully-developed turbulent channel flow (TCF) case. To relax the additional time step restrictions, the modified Discontinuous Galerkin (mDG) approach from Krivodonova et al is employed. The mDG approach allows larger time steps, but with a trade-off in terms of accuracy. We show how mDG can be combined with p-adaptivity to relax the time step restrictions. This is first done for the TGV case using dynamic p-adaptation. There it is shown how combining mDG with p-adaptation results in lower number of time steps. Next, the TCF is studied using a static "p-map", that is, higher p at the wall adjoining elements and lower p everywhere else. It will be shown that this results in improved statistics without having the usual wall restriction of placing the first point within y+ = 1. It is further shown how mDG can relax the time step restrictions required because of the refinement for this case.

Minisymposium Organisers

Thomas Hagstrom, Southern Methodist University
Oscar Bruno, California Institute of Technology

Abstract

With increases in computing power, a goal for computational science is to solve more complex and challenging physical problems. In wave theory the fundamental issues to be addressed include:

Minimization of dispersion/dissipation errors for problems with hundreds or more wavelengths and thousands of periods in a given simulation

Accurate representation of complex geometrical features

Efficient quantification of the effects of uncertainties in the problem specification

In simple domains, it is easy to show that high-order/high-resolution methods maximize computational efficiency as the number of wavelengths over which a wave must be accurately propagates becomes large, with Fourier methods for periodic problems being the gold standard. Speakers in this minisymposium will discuss a variety of novel approaches to to achieve the promise of high-order space-time discretizations in realistic settings. Ideas include the extension of Fourier methods to non-periodic problems, new time-stepping strategies beyond the standard method-of-lines, robust discretizations on overlapping structured grids, the consequences of high-order approximations for uncertainty quantification, and approximations to new time-stable formulations of time-domain integral equations.

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Authors (presenter in bold)

William D. Henshaw, Rensselaer Polytechnic Institute
Jeffrey W. Banks, Rensselaer Polytechnic Institute
Jordan B. Angel, Rensselaer Polytechnic Institute

Abstract

In this talk I will describe the development of high-order accurate finite-difference based upwind-schemes for the second-order wave equation (e.g. Maxwell's equations in second-order form) on complex geometries using composite overlapping grids. The approach uses the recent work of Banks and Henshaw that showed how to incorporate upwinding directly into the second-order wave equation by using the d'Alembert solution for an appropriate Riemann problem. Traditional non-dissipative high-order schemes using centered finite-differences for the wave equation on overlapping grids are usually unstable due to an interaction between the interpolation points and nearby boundaries. It is found through normal mode analysis and practical computations that the new high-order upwind schemes are robust and stable on overlapping grids even in the difficult case of a thin boundary layer grid when the interpolation points lie close to a physical boundary. Properties of the new scheme are demonstrated through two and three dimensional solutions to Maxwell's equations. An optimized form of the upwind scheme is found to have a large "CFL one" time-step and to be very efficient especially when most of the domain is covered with a background Cartesian grid.

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Authors (presenter in bold)

Oscar P. Bruno, Caltech

Abstract

We present fast spectral solvers for time-domain Partial Differential Equations. Based on a novel Fourier-Continuation (FC) method for the resolution of the Gibbs phenomenon, these methodologies give rise to time-domain solvers for PDEs for general engineering problems and structures which enjoy a number of desirable properties, including spectral time evolution essentially free of pollution or dispersion errors for general PDEs in the time domain, with conditional/unconditional stability for explicit/alternating-direction methods and high order of temporal accuracy. In particular, novel BDF-based time stepping methods of orders of accuracy two to six will be described which, without recourse to solution of nonlinear systems, can lead to highly-stable algorithms: the new second-order scheme is rigorously shown to be unconditionally stable (for linear problems), and the higher schemes are shown to be "quasi-unconditionally stable" -- a stability concept, close to but different from unconditional stability, which will be defined as part of this presentation. A variety of applications to linear and nonlinear PDE problems will be presented, including the diffusion and wave equations, the Navier-Stokes equations and the elastic wave equation, demonstrating the significant improvements the new algorithms can provide over the accuracy and speed resulting from other approaches.

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Authors (presenter in bold)

Jay Gopalakrishnan, Portland State University

Abstract

Solutions of hyperbolic systems have finite speed of propagation. This leads us to wonder if tent-shaped domains are more natural for solving hyperbolic systems. A spacetime simulation region can be subdivided into tent-shaped subregions. Within each tent, we can ensure causality by constraining the height of the tent pole. With this as the starting point, we introduce new schemes, called Mapped Tent Pitching (MTP) schemes, which compute a numerical solution of a hyperbolic system on spacetime advancing fronts made of tent canopies. Locally variable time step sizes, even on unstructured meshes, are natural in this approach. The new mapping techniques allow, for the first time, to use fully explicit schemes within tents. These degenerate maps transform tents into domains where space and time are separated, thus allowing, also for the first time, standard methods to be used within tents. Several open mathematical and computational issues surrounding these methods will be touched upon.

Reference: J. Gopalakrishnan, J. Schöberl, and C. Wintersteiger. Mapped tent pitching schemes for hyperbolic systems. SIAM J Sci Comp, 39:6, p.B1043-B1063, 2017.

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Authors (presenter in bold)

Thomas G. Anderson, California Institute of Technology
Oscar P. Bruno, California Institute of Technology
Mark Lyon, University of New Hampshire

Abstract

We present a new algorithm for the solution of time-domain wave propagation problems in unbounded domains. Unlike existing solution methods for this problem, including finite-difference, finite-element and convolution-quadrature methods, the proposed approach does not rely on time stepping. Instead, the new method is based on use of a continuous Fourier Transform in time, to decouple individual frequency problems which can then solved by efficient boundary integral methods. Importantly, the algorithm produces the solution at any prescribed point in time without requiring evaluation of the solution at any prior times. Utilizing a combination of the recently-introduced windowed Green function method in conjunction with suitable transformations between time-domain and frequency-domain integral equations, high-order numerical algorithms for frequency-domain solution, Fourier-transforms and O(1) high-frequency integration, the entire methodology is highly efficient and accurate, as well as capable of embarrassingly parallel implementations, for general scattering problems in two and three dimensional space.

Minisymposium Organisers

Nail Yamaleev, Old Dominion University
Mark H. Carpenter, NASA Langley Research Center

Abstract

Although various nonlinearly stable methods based on 1st and 2nd-order discretizations are available in the literature (e.g., TVD, TVB, entropy stable schemes, etc.), extension of these techniques to high-order formulations has been uncommon. The main objective of this minisymposium is to present and discuss new approaches and future directions for constructing high-order methods that mimic the key stability properties of governing nonlinear partial differential equations including entropy stability, dissipation of kinetic energy, and others.

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Authors (presenter in bold)

Magnus Svärd, University of Bergen

Abstract

A proof of convergence for a numerical scheme is also a proof of well-posedness. Hence, such a proof is very unlikely given that there is no well-posedness proof obtained by any method for the Navier-Stokes equations. Nevertheless, a convergence proof, preferably for a high-order accurate scheme, is what CFD needs in order to produce credible predictions.

For subsonic flows, linear stability for schemes of arbitrary order, is generally considered to be sufficient for convergence in cases when the solution can be assumed to be smooth. In the talk, I will make some comments regarding subsonic flows. However, the main topic is the general case when a solution is not a priori assumed to exist. In any convergence theory, a priori estimates are necessary and there are different strategies to achieve such estimates. A common, and promising technique for numerical schemes, is entropy stability which is what I will base my talk on. However, entropy stability is far from sufficient and estimates on the velocity gradients appear to be necessary. Furthermore, a scheme must also satisfy positivity constraints.

The question is: Is it possible to satisfy the suite of estimates and constraints while still producing high-order accurate approximations to smooth solutions?

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Authors (presenter in bold)

Jan Nordström, Linköping University

Abstract

We derive energy bounding boundary conditions for nonlinear problems by using the energy method. We discuss in particular, the necessity to sometimes use nonlinear boundary conditions, different formats for the boundary conditions and its relation to a few commonly used ones. We also point out the main differences compared with linear analysis.

By mimicking the continuous analysis, we prove stability of the semi-discrete approximation using SBP-SAT in space. Finally, we show that the semi-discrete stability results leads directly to fully discrete stability, by using SBP-SAT also in time. The discrete analysis is carried out for an approximation on tensor product form, but is valid for all types of approximations on SBP-SAT form.

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Authors (presenter in bold)

Jared Crean, Rensselaer Polytechnic Institute
David C. Del Rey Fernandez, National Institute of Aerospace
Mark H. Carpenter, NASA Langley Research Center
Jason E. Hicken, Rensselaer Polytechnic Institute

Abstract

This presentation will describe ongoing work to extend entropy stability to Summation-by-Parts (SBP) discretizations on staggered, unstructured meshes. Entropy stability provides a way to show the stability of nonlinear problems, and is particularly important for high-order discretization which have low numerical viscosity. Recent work by Fisher and Carpenter (DOI:10.1016/j.jcp.2013.06.014) have enabled the construction of SBP discretizations that are entropy stable. This work was extended to SBP discretizations on unstructured grids (DOI:10.1016/j.jcp.2017.12.015, DIO:10.1016/j.jcp.2017.05.025). For tensor product elements, Parsani et al. have developed a staggered approach (DOI:10.1137/15M1043510), which we seek to extend to unstructured grids.

For under-resolved flow simulations, implicit time marching methods are often used to avoid excessive CFL restrictions. Using a staggered formulation, the size of the matrix that must be inverted is reduced while retaining favorable properties of the flux grid operator. For example, the size of the matrix is minimized by choosing a solution grid SBP operator with the minimum number of nodes. Computing the entropy-conservative face integrals for these operators requires a large number of flux evaluations due to the dense interpolation operator between the volume nodes and the face cubature points. To avoid this expense, an SBP operator with a diagonal boundary operator is advantageous because it reduces the number of flux function evaluations required. By choosing the minimum node operator for the solution grid and the diagonal boundary operator for the flux grid, a staggered formulation obtains the benefits of both operators simultaneously.

After reviewing these formulations briefly, we will present a new formulation that uses different sets of nodes for the solution and flux. Particular emphasis will be placed on the properties of the interpolation operators that relate the solution and residual on the different node sets. These properties are essential to proving the entropy stability of the staggered scheme.

Preliminary numerical results of the Euler equations will verify the entropy stability of the scheme for a periodic problem and demonstrate accuracy with a convergence study on the unsteady vortex.

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Authors (presenter in bold)

Nail Yamaleev, Old Dominion University

Abstract

In contrast to the existing shock-capturing methods that introduce dissipation based on smoothness of a discrete solution, we propose to regularize the compressible Navier-Stokes equations by adding artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. This regularization satisfies the first and second law of thermodynamics, ensures positivity of thermodynamic variables, and preserves the translational and rotational invariance at the continuous level. In this talk, we present a new class of artificial dissipation spectral collocation operators of arbitrary order of accuracy, that mimic some key properties of the continuous Brenner-Navier-Stokes diffusion operator at the discrete level. The new artificial dissipation spectral collocation operators preserve the superconvergence properties of the baseline spectral collocation operators, satisfy the summation-by-parts convention and discrete entropy inequality, thus facilitating a nonlinear L2-stability proof for the symmetric form of the regularized Navier-Stokes equations. Numerical results demonstrating accuracy and non-oscillatory properties of the new schemes are presented for both continuous and discontinuous flows.

Minisymposium Organisers

Stefan Sauter, University of Zurich
Markus Melenk, TU Vienna

Abstract

High-frequency scattering problems have many important applications which include, e.g., radar and sonar detection as well as medical and seismic imaging. Their efficient and reliable numerical modelling is a challenge and the development of fast numerical methods is a topic of vivid worldwide research in numerical analysis and scientific computing. Such problems are often modelled in the frequency domain, where a time-periodic ansatz is employed for the wave or Maxwell equation which, in turn, results in the Helmholtz or stationary Maxwell equation in the high-frequency regime, i.e., with large wave number. High order methods are particularly suited for this problem class. The aim of this minisymposium is to bring together leading experts in the area of the numerical analysis and scientific computing in this area.

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Authors (presenter in bold)

Bjorn Engquist, University of Texas, Austin
Hongkai Zhao, UC Irvine

Abstract

The Helmholtz equation is a linear partial differential equation modeling time harmonic wave propagation. It is notoriously difficult to solve numerically when the frequency is high. I will present our recent study on the intrinsic complexity of the Helmholtz operator in the high frequency limit. The study is based on the analysis of approximate separable representations of the Green’s functions. Computationally, being able to approximate a Green’s function by a sum with few separable terms is equivalent to the existence of low rank approximation of the discretized systems which can be explored for matrix compression and fast solution techniques. We prove both lower and upper bounds for the number of terms needed for a separable approximation of the Green’s function of the Helmholtz operator. Sharpness and implications of these bounds will be shown for computation setups that are commonly used in practice.

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Authors (presenter in bold)

Ralf Hiptmair, ETH Zurich
Daniele Casati, ETH Zurich

Abstract

We consider scalar and electromagnetic wave propagation in frequency domain on un- bounded domains, partly filled with inhomogeneous media. We propose a discretization relying on Trefftz approximation in an unbouded exterior domain through fields generated by multipole auxiliary sources, combined with conventional mesh-based primal Lagrangian finite elements in other parts.

We develop and, based on variational saddle point theory, analyze several approaches for coupling both discretizations across interfaces. Thus we obtain quantitative asymptotic convergence results that suggest optimal ways to balance the resolutions of Trefftz spaces and finite elements. In particular, it is highly advisable to ensure analyticity of the solution beyond the coupling interface, which, therefore, should be an artificial interface at a safe distance from discontinuities in the material properties.

By means of numerical ecperiments we compare the performance of different coupling schemes.

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Authors (presenter in bold)

Ivan Graham, University of Bath
Euan Spence, University of Bath

Abstract

There is currently large research interest in finding optimal-in-time solvers for finite-element discretisations of frequency-domain wave problems, such as the Helmholtz and time-harmonic Maxwell equations, when the frequency is large. Ideally such solvers should also have good parallel scaling properties, be robust to heterogeneities in the material coefficients. and come with theorems rigorously justifying their behaviour.

A common approach to this problem is trying to find good preconditioners to use when solving the linear systems with (F)GMRES.

This talk will be about preconditioners built using (i) variants of classical additive-Schwarz domain-decomposition methods, and (ii) artificial absorption (similar to in the "shifted Laplacian" preconditioner involving multigrid).

The overall philosophy is to use PDE theory of the underlying boundary-value problems to tackle this linear-algebra problem of developing fast solvers.

The work on Helmholtz is joint with Eero Vainikko (Tartu), and Jun Zou (Chinese University of Hong Kong).

The work on Maxwell is joint with Marcella Bonazzoli (Paris 6), Victorita Dolean (Strathclyde/Cote d'Azur), and Pierre-Henri Tournier (Paris 6).

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Authors (presenter in bold)

Théophile Chaumont-Frelet, BCAM
Serge Nicaise, Université de Valenciennes

Abstract

The solution to the Helmhotlz equation can exhibit singularities close to the corners of the boundary when it is set in non-convex domains. An important example is the scattering of a wave by a polygon: in this case, the solution features singularities close to the corners of the scatterer. It is especially important to analyze these singularities, as they heavily impact numerical discretization techniques.

In this work, we focus on acoustic Helmholtz problems set in 2D. In a first part, we characterize the singularities of the solution close to boundary corners. This is done by splitting the solution into a regular part, and one singular function for each corner. Then, we give sharp estimates that describe the behavior of the regular part and each singular function as the frequency grows.

In a second part, we take advantage of the aforementioned estimates to analyze finite element discretizations. Specifically, we focus on first-order Lagrange finite element and we derive asymptotic and pre-asymptotic error estimates that generalize previous results obtained in smooth domains. Our main conclusion is that the so-called "pollution effect" is not affected by the presence of singularities. In addition, for high frequency problems, our error estimates show that unless an especially fine accuracy level is required, there is no need to refine the finite element mesh close to the singular corners.

In a third part, we provide numerical experiments that illustrate our error analysis. These test-cases further confirm our observations concerning the "pollution effect" and local mesh refinements.

Minisymposium Organisers

Jae-Hun Jung, SUNY Buffalo, Ajou University
Sigal Gottlieb, UMass Dartmouth

Abstract

Data analysis such as handling, interpretation, and utilization of data emerges in recent years as a crucial component in most research areas of science and humanity. The proposed mini-symposium attempts to see how research in high order methods could play a role in the era of data science. This mini-symposium will bring together researchers in the high order research community whose current research finds data analysis a critical part of their research. It will also provide a unique opportunity for researchers in high order methods area to find a possible research agenda in data science by sharing their expertise and experiences. In this mini-symposium recent progress of data analysis methods in dealing with data created by high order methods will be introduced. The mini-symposium will also present recent works of deep learning, topological data analysis of spectral approximations and their applications.

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Authors (presenter in bold)

Scott Field, University of Massachusetts Dartmouth

Abstract

Parameterized inner products which must be repeatedly computed are encountered in many areas such as Bayesian data analysis. For example, in Bayesian parameter estimation, an experimentally recorded dataset is compared to a parameterized model. For noisy data taken at equally spaced intervals, the relevant likelihood computations are, essentially, inner products evaluated by a low-order Riemann sum. This talk presents an algorithm to generate reduced order quadratures (ROQs) for multiple fast evaluations of inner products between parameterized functions/models and noisy data. Generation of an ROQ rule is carried out offline. First, a projection-based greedy algorithm identifies a basis whose span accurately approximates the model. An empirical interpolation procedure is then applied to select ROQ nodes. The resulting quadrature rule's computational cost depends linearly on the number of ROQ nodes and, furthermore, for smooth models, ROQs are shown to converge exponentially fast with the number of nodes thereby significantly reducing the overall time to carry out a parameter estimation study. Examples will draw from applications of this method to gravitational waves (GW) physics including its use in the most recent set of GW observations made by LIGO.

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Authors (presenter in bold)

Myungjoo Kang, Seoul National University
Seongju Do, Seoul National University
Youngsoo Ha, Seoul National University
Changho Kim, Konkuk University

Abstract

We present nonoscillatory semi-discrete central-upwind schemes with a modified multidimensional limiting process (MLP) for solving the equations of compressible ideal magnetohydrodynamics (MHD) in multiple space dimensions. This approach is based on directly solving the equations with a new limiting strategy to control the oscillations at nonsmooth regions and to yield decreasing numerical dissipation. The central-upwind scheme retains the simplicity of the Lax-Friedrichs scheme, which has no Riemann solvers and no local characteristic decompositions. The proposed scheme employs the central-upwind scheme in semi-discrete form combined with the constrained transport technique to satisfy the magnetic divergence-free condition for the magnetic field. A variety of test problems are provided to demonstrate the ability of the proposed scheme.

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Authors (presenter in bold)

Misun Min, Argonne National Laboraotry

Abstract

We will present very interesting and challenging ion channel problems for studing the dynamics of ion concentration within complicated protein structure that is represented by point data with charge distribution.

This involves solving drift-diffusion models describing the transport of the charge-carriers coupled with a Poisson equation for electric potential. The spatial discretization is based on a standard spectral-element formulation with body-fitted hexahedral elements. The temporal discretization is a mixed implicit-explicit method using kth-order backward-difference formulas and extralpolation. Spectral element multigrid methods are considered for solving the electric potential.

The detailed treatment representing the material properties of the data on the high-order spectal-element meshes and the PDE models and their discretizations for representing the ion size effect will be discussed.

Minisymposium Organisers

Misun Min, Argonne National Laboratory
Paul Fischer, University of Illinois, Argonne National Laboratory
Tzanio Kolev, Lawrence Livermore National Laboratory

Abstract

High-order discretizations have the potential to provide an optimal strategy for achieving high performance and delivering fast, efficient, and accurate simulations on next-generation architectures. Traditional low-order schemes, developed in an era when FLOPs were the performance-limiting factor, will no longer be an attractive option, since the cost of FLOPs will be considerably smaller than the cost of data motion. While many high-order methods were intractable on older architectures, their importance and usefulness has been dramatically increasing and has demonstrated significant improvements in both simulation quality and parallel strong scalability.

This session will discuss the next-generation high-order discretization algorithm and software, based on finite/spectral element approaches that will enable a wide range of important scientific applications to run efficiently on future architecture.

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Authors (presenter in bold)

Tzanio Kolev, Lawrence Livermore National laboratory

Abstract

Efficient exploitation of exascale architectures requires rethinking of the numerical algorithms used in many large-scale applications. These architectures favor algorithms that expose ultra fine-grain parallelism and maximize the ratio of floating point operations to energy intensive data movement. One of the few viable approaches to achieve high efficiency in the area of PDE discretizations on unstructured grids is to use matrix-free/partially-assembled high-order finite element methods, since these methods can increase the accuracy and/or lower the computational time due to reduced data motion.

In this talk we report on recent work in the Center for Efficient Exascale Discretizations (CEED), a co-design center in the US Exascale Computing Project that is focused on the development of next-generation discretization software and algorithms to enable a wide range of finite element applications to run efficiently on future hardware. CEED is a research partnership involving 30+ computational scientists from two US national labs and five universities, including members of the Nek5000, MFEM, MAGMA, OCCA and PETSc projects.

Topics of discussion will include recent progress in CEED packages and applications, new miniapps, benchmarks and API libraries developed in the project, and our efforts in scalable unstructured adaptive mesh refinement, matrix-free linear solvers and high-order data analysis and visualization.

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Authors (presenter in bold)

Paul Fischer, University of Illinois

Abstract

We describe several recent advances in the development of the spectral element method (SEM) [1] for scalable large-eddy simulations (LES) of turbulent flows in complex domains. The SEM yields minimal numerical dispersion for the resolved scales, which results in an order of magnitude reduction in the number of gridpoints required for accurate simulation of turbulence when compared with traditional 2nd-order schemes. Low cost for the SEM is realized through fast tensor-product operator evaluation coupled with scalable multilevel preconditioners for the elliptic (i.e., viscous/pressure) substeps in the time advancement of Navier-Stokes equations (NSE).

New advances include the development of nonconforming overlapping Schwarz methods for the NSE, characteristics-based timesteppers for arbitrary Lagrangian-Eulerian formulations, nonlinear filter-based stabilization, scalable AMG-based preconditioners and fast multi-way Lagrangian particle tracking.

The Schwarz method is a particularly exciting development in that it greatly increases geometric and multiphysics flexibility. The implementation exploits fast and scalable general interpolation routines in the gslib communication library, which is part of Nek5000. The Schwarz methods overcome significant challenges posed when adjacent subdomains have complex geometry/topology constraints that make conformal mesh generation a challenge. We discuss interface condition requirements, stability, and accuracy of the Schwarz methods as applied in these contexts.

We present scaling results (out to a million processes) and turbulence simulation results throughout the presentation.

[1] Patera, A.T., J.Comput.Phys.54,468(1984)

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Authors (presenter in bold)

Veselin Dobrev, Lawrence Livermore National Laboratory

Abstract

In this talk, we present an overview of the modular finite element methods (MFEM) library (mfem.org), including its main abstraction classes, their corresponding linear algebra objects, and implementation variants. We discuss the components required for the construction and application of general finite element discretization operators and the various choices for their software implementation, e.g. as a single assembled parallel CSR matrix, or as a product of linear operators, i.e. "matrix-free" representations. We highlight the pros and cons of the various choices based on the discretization parameters such as solution space order, mesh order, choice of quadrature, etc., and present numerical illustration with MFEM examples. We also report on the progress of the ongoing efforts to implement efficiently and integrate seamlessly support for architectures with accelerators (e.g. GPUs) in the library. The behavior of the various algorithms with respect to scalability on HPC systems is also studied numerically and discussed.

Prepared by LLNL under Contract DE-AC52-07NA27344. LLNL-ABS-745578.

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Authors (presenter in bold)

Peter E. Vincent, Imperial College London
Arvind S. Iyer, Imperial College London
Freddie D. Witherden, Stanford University
Brian C. Vermeire, Concordia University
Yoshiaki Abe, Imperial College London
Ralf-Dietmar Baier, MTU Aero Engines AG
Antony Jameson, Stanford University

Abstract

The turbine stages of a jet engine extract energy from exhaust gasses to drive the compressor, fan, and other auxiliary systems. Approximately half the weight of a turbine stage comes from the turbine blades. In order to reduce this contribution, modern turbines are designed to use as few blades as possible. However, this results in individual blades being under higher-loading, which can lead to fully-separated flow over the aft-portion of each blade; introducing complex, unsteady, three-dimensional phenomena, and thus reduced LPT efficiency, which can have a significant negative effect on overall fuel consumption rate. Accurate simulation of unsteady turbulent flow in the vicinity of complex geometric configurations is critical for improved design of turbine stages, and hence ‘greener’ aircraft that are more fuel-efficient. In this talk we will demonstrate application of PyFR [1], a high-order accurate Python based computational fluid dynamics solver, to petascale simulation of flow over low-pressure turbine linear cascades. Rationale behind algorithmic choices, which offer increased levels of accuracy and enable sustained computation at up to 58% of peak DP-FLOP/s on unstructured grids, will be discussed in the context of modern hardware. A range of software innovations will also be detailed, including use of runtime code generation, which enables PyFR to efficiently target multiple platforms, including heterogeneous systems, via a single implementation.

[1] F. D. Witherden, A. M. Farrington, P. E. Vincent. PyFR: An Open Source Framework for Solving Advection-Diffusion Type Problems on Streaming Architectures using the Flux Reconstruction Approach. Computer Physics Communications, Volume 185, Issue 11, Pages 3028-3040, 2014.

Minisymposium Organisers

Bryan Quaife, Florida State University
Travis Askham, University of Washington
Ludvig af Klinteberg, Simon Fraser University

Abstract

In the modern computing environment, high-order methods are of increasing importance for the accurate simulation of complex fluid flows. Applications include porous media flow, suspensions of rigid and deformable particles, and multiphase flows. Numerical simulations of such complex systems are challenging because of strong nonlinearities, non-local interactions, time-dependent geometries, multiple length and time scales, and fluid-structure interactions. Integral equation methods offer a powerful and flexible framework that addresses many of these challenges while providing high-order accuracy. This symposium will discuss recent advances in high-fidelity and scalable integral equation methods in fluid dynamics.

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Authors (presenter in bold)

Travis Askham, University of Washington

Abstract

In this talk, we briefly outline an integral equation method for the simulation of the Navier-Stokes equations in the plane. Such a method requires fast sums for many derivatives of the Green's function for the so-called modified biharmonic equation. We explain some numerical difficulties associated with creating an analytic fast multipole method for these interactions and describe a new set of special functions which make the method stable.

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Authors (presenter in bold)

Shilpa Khatri, University of California, Merced

Abstract

Accurate evaluation of layer potentials near boundaries and interfaces are needed in many applications, including fluid-structure interaction problems. A classical method to approximate the solution everywhere in the domain consists of using the same quadrature rule (Nyström method) used to solve the underlying boundary integral equation. This method is problematic for evaluations close to boundaries and interfaces. For a fixed number, N, of quadrature points, this method incurs a non-uniform error with O(1) errors in a boundary layer of thickness O(1/N). We have developed a new method to remove this O(1) error without having to use high-order Nyström methods. To demonstrate this method, we consider the interior and exterior Laplace problems.

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Authors (presenter in bold)

Alexander Heinecke, Intel Corporation
Alexander Breuer, UC San Diego

Abstract

In recent years the compute/memory balance of processors has been continuously shifting towards compute. Especially the rise of Deep Learning, which is based on matrix multiplication (GEMM), accelerated this path, especially in terms of single precision and lower precision compute. An important research question is if this development can be leveraged for traditional HPC. Therefore the charter of this talk two-fold: first we will give a general overview why high order methods are an excellent match to modern, unbalanced hardware architectures as these solvers do not suffer from extremely high flop to byte ratios. As flops are free, high order methods can now unfold their full potential with respect to time-to-solution, or even more important energy-to-accuracy. Here, we will focus on spectral element CFD applications and discontinuous Galerkin FEM for solving seismic wave equations. Based on these learnings, the second part proposes further extensions to solvers to GEMM-optimized hardware by taking the Intel Xeon Phi x205 family (code-named Knights Mill) optimized for deep learning as an example. Efficiently utilizing deep learning hardware has two major challenges: a) it has to be proven that the solver of interest can be run in single precision or even lower precision (Knights Mill provides 14 TFLOPS single precision performance at 460 GB/s bandwidth per socket) and b) efficient mapping on-to GEMM-hardware requires rethinking. Efficient exploitation of these types of hardware are possible by taking the entire simulation pipeline into consideration and not limiting potentially optimization to a traditional solver approach. Orthogonal to a raw performance (flops and accuracy) evaluation, we will also summarize the software engineering techniques used. Which included, but are not limited, to careful software design having hardware characteristics in mind and automatic code generation. Latter should ideally happen at runtime in a just-in-time fashion to maintain best possible flexibility of the solver. Finally, we will generalize proposed methods and provide an outlook towards upcoming architectures and their fitness for high order methods.

Minisymposium Organisers

Francisco Pla, Departamento de Matemáticas, Facultad de Ciencias y Tecnologías Químicas, Universidad de Castilla-La Mancha, 13071 Ciudad Real

Abstract

The main objective of this minisymposium is to present some recent advances in high-order and spectral methods for flows applied to geophysical models in land, air or sea, where the high-order and spectral methods play a relevant role due to these are capable of producing very accurate numerical results of those differential equations that model geophysical problems.

Different numerical tecniques and computing implementations will be presented, such as degree adaptation techniques for high-order, discontinuous Galerkin space discretization, parallel implementations and direct numerical simulation (DNS).

This minisymposium will bring together scientists to present their recent advances on algorithm development and analysis of high-order and spectral methods applied to geophysical issues, provide a forum for discussion and interaction in these methods.

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Authors (presenter in bold)

Jezabel Curbelo, Universidad Autónoma de Madrid
Ana M. Mancho, Instituto de Ciencias Matemáticas

Abstract

We explore the instabilities developed in a fluid in which viscosity depends on temperature. For these reason, a spectral numerical method is proposed to solve the time evolution of a convection problem in a 2D domain at infinite Prandtl number. We have considered periodic boundary conditions along the horizontal coordinate which introduce the O(2) symmetry into the setting. The analysis is assisted by bifurcation techniques such as branch continuation, which has proven to be a useful, and systematic method for gaining insight into the possible stationary solutions satisfied by the basic equations.

Several viscosity laws which correspond to different dependences of the viscosity with the temperature are investigated. Numerous examples are found along the branching diagrams, in which stable stationary solutions become unstable through a Hopf bifurcation. In the neighborhood of these bifurcation points, the scope of our techniques is examined by exploring transitions from stationary regimes towards time dependent regimes. We also compare the output and performance of these methods with other schemes.

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Authors (presenter in bold)

Luca Bonaventura, Politecnico di Milano
Giovanni Tumolo, OGS-ICTP

Abstract

Static and dynamic degree adaptation techniques for high order, discontinuous Galerkin space discretizations will be introduced, that allow to reduce substantially the number of degrees of freedom employed in large scale simulations. The main features of a parallel implementation presently being developed will also be reviewed. Numerical results on model problems relevant for numerical climate and weather prediction, oceanic modelling, large eddy simulation will be presented. The work presented is a joint development with G. Tumolo, ICTP and OGS, Trieste, Italy.

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Authors (presenter in bold)

María Cruz Navarro, Universidad de Castilla- La Mancha
Damián Castaño, Universidad de Castilla- La Mancha
Henar Herrero, Universidad de Castilla- La Mancha

Abstract

In this paper we use nonlinear simulations to show the generation of single-cell vortices, two-cell vortices and double vortices in a cylinder non-homogeneously heated from below in a rotating reference frame. For moderate rotation rates we show the transition from an axisymmetric one-cell vortex, characterized by an updraft at the center of the cell, to an axisymmetric two-cell vortex, characterized by a central downdraft surrounded by updraft, i.e., a vortex that develops a central eye. This transition can be explained through a force balance analysis. When the thermal gradient increases beyond a certain threshold the axisymmetric single-eyed vortex loses the axisymmetry, the eye displaces from the center and tilts. For larger rotation rates the axisymmetric one-cell vortex bifurcates to a double vortex.

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Authors (presenter in bold)

Giovanni Tumolo, OGS-ICTP
Luca Bonaventura, Politecnico di Milano

Abstract

High order DG methods pose stringent stability restrictions on explicit time discretization methods. In recent work by the authors, a strategy for the reduction of the computational cost of DG methods has been proposed, based on a novel semi-implicit, semi-Lagrangian time discretization and on a dynamically adaptive choice of the polynomial degree employed in each element which allows to reduce the number of degrees of freedom by a factor of up to 50%. In this work, we will outline the ongoing development of a parallel library that will allow to implement these adaptive techniques by taking full advantage of degree adaptivity in each coordinate direction of generic hexaedral elements. The preliminary results achieved are promising and hint that the proposed approach is useful to achieve the maximum possible flexibility in the application of high order discontinuous finite element methods to environmental flows.

Minisymposium Organisers

Thomas Hagstrom, Southern Methodist University
Oscar Bruno, California Institute of Technology

Abstract

With increases in computing power, a goal for computational science is to solve more complex and challenging physical problems. In wave theory the fundamental issues to be addressed include:

Minimization of dispersion/dissipation errors for problems with hundreds or more wavelengths and thousands of periods in a given simulation

Accurate representation of complex geometrical features

Efficient quantification of the effects of uncertainties in the problem specification

In simple domains, it is easy to show that high-order/high-resolution methods maximize computational efficiency as the number of wavelengths over which a wave must be accurately propagates becomes large, with Fourier methods for periodic problems being the gold standard. Speakers in this minisymposium will discuss a variety of novel approaches to to achieve the promise of high-order space-time discretizations in realistic settings. Ideas include the extension of Fourier methods to non-periodic problems, new time-stepping strategies beyond the standard method-of-lines, robust discretizations on overlapping structured grids, the consequences of high-order approximations for uncertainty quantification, and approximations to new time-stable formulations of time-domain integral equations.

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Authors (presenter in bold)

Thomas Hagstrom, Southern Methodist University
Alex Barnett, Flatiron Institute
Charles Epstein, University of Pennsylvania
Leslie Greengard, Flatiron Institute

Abstract

We consider the formulation and high-order discretization of time-domain integral equations for the scattering of acoustic waves. Through the detailed study of the case of the round sphere the long time stability of the solution for various combined field formulations is revealed. Extending these to convex scatterers, we propose a formulation for which the time decay of the retarded potential matches the time decay of the dominant scattering resonances. As such the problem of recovering the acoustic field from the potential does not become exponentially ill-conditioned as time increases. In contrast, the retarded single or double layer potentials alone are dominated by internal resonances as time increases while the wave field itself decays exponentially, so the recovery problem becomes increasingly difficult to solve.

To discretize the system we use high-order quadrature in space and march in time using a predictor-corrector method based on what we call difference splines. Unlike previously proposed spline-based marching algorithms, difference splines are based on interpolation rather than quasi-interpolation and can achieve arbitrary order. However, as with other spline-based algorithms, and in contrast with convolutional quadrature methods, they retain the finite time history inherent in the problem. Experiments with the proposed discretization are presented for scattering by a torus. Although this geometry is not covered by the theory developed for convex scatterers, we find that the combined field formulation works well. We also find that the fully implicit time marching scheme defined by the difference splines is stable subject to an inverse CFL-type restriction - that is so long as the time step is not too small relative to the spatial mesh parameters. The predictor-corrector scheme is also found to be stable so long as sufficiently many corrector iterations are used. As such the method is explicit in the sense that the solution update at any point only depends on the updated solution at neighboring points.

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Authors (presenter in bold)

Daniel Appelo, University of Colorado Boulder
Thomas Hagstom, Southern Methodist University
Mohammad Motamed, University of New Mexico
Arturo Vargas, Lawrence Livermore Natl. Labs.
Yunan Yang, University of Texas at Austin

Abstract

In this talk we report on the development of high order Hermite and dG methods for wave equations in second order form and its applications to forward propagation of uncertainties and the inversion of material parameters from boundary measurements.

The Hermite methods come in two flavors, dissipative and conservative. The dissipative method achieves space-time accuracy of order (2m+1), while the conservative method has space-time order (2m). Besides their high order of accuracy in both space and time combined, the methods have the special feature that they are stable for CFL numbers up to and including unity for all orders of accuracy. Curiously, when using CFL equal to one, the orders of both methods increase.

Our spatial discontinuous Galerkin discretization of wave equations in second order form relies on a new energy based strategy featuring a direct, mesh-independent approach to defining interelement fluxes. Both energy-conserving and upwind discretizations can be devised. The method comes with optimal a priori error estimates in the energy norm for certain fluxes and we present numerical experiments showing that optimal convergence for certain fluxes.

The forward propagation of uncertainties is performed by a new multi-order Monte Carlo algorithm for computing the statistics of stochastic quantities of interest. The method is a non-intrusive technique based on a high (but variable) order discretizations, as those presented. The algorithm is built upon a hierarchy of degrees of polynomial basis functions rather than a mesh hierarchy used in multi-level Monte Carlo. In addition to the convenience of working with a fixed mesh, which is desirable in many real applications with complex geometries, the multi-order method is particularly beneficial in reducing errors due to numerical dispersion in long-distance propagation of waves.

To invert for P and S velocities in the elastic wave equation we extend the novel trace-by-trace quadratic Wasserstein metric (W2) technique pioneered by Yang and co-authors. We give examples comparing W2 with the standard L2 misfit.

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Authors (presenter in bold)

Marc Gerritsma, TU Delft
Yi Zhang, TU Delft
Varun Jain, TU Delft
Artur Palha, TU Eindhoven

Abstract

Conservation at the continuous level does not necessarily imply that physical invariants are also conserved in a finite dimensional setting. Ideally, one would like to conserve important physical properties (mass, kinetic energy, helicity, etc) independent of the shape of the mesh (orthogonal or curved grids), the size of the mesh (a coarse grid or very fine mesh) and the polynomial approximation (low order methods vs high order methods).

In [1] conservation of the time-dependent 2D Navier-Stokes equations were considered with periodic boundary conditions. This paper shows that mass and vorticity are exactly conserved at the discrete level. In the limit of vanishing viscosity also the total kinetic energy and helicity are conserved in time.

In [2] these ideas are extended to the shallow water equation and in [3] the inclusion of no-slip boundary conditions is discussed.

Conservation -- independent of size, shape or order -- can be accomplished by a careful selection of degrees of freedom in space and time. In the time domain the finite dimensional function spaces should form a De Rham complex and in time a staggered time-stepping approach was employed.

In this talk these ideas will be applied to wave equations. Here we will employ algebraic dual spaces, [4], in space and time to ensure conservation. Conservation will be proved and results will be presented.

[1] Palha and Gerritsma, A mass, ebergy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations, JCP 328, pp. 200-220, 2017.

[2] Lee, Palha and Gerritsma, Discrete conservation properties for shallow water flows using mixed mimetic spectral elements, JCP 357, pp. 282-404, 2018.

[3] Gonzalez de Diego, Palha and Gerritsma, Inclusion of no-slip boundary conditions in the MEEVC scheme , submiited to JCP, 2017.

[4] Jain, Zhang, Palha and Gerritsma, Construction and application of algebraic dual polynomial representations for finite element methods, arXiv:1712.09472, 2017.

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Authors (presenter in bold)

Rodrigo Platte, Arizona State University

Abstract

Windowed Fourier transforms have long been used to study local features of signals and are often called short-time Fourier transforms. They are commonly used to determine frequency and phase content of local sections of a signal as it changes over time. They are, however, rarely used in the solution and analysis of differential equations because regions near the boundaries of the domain are difficult to handle. Smooth manifolds, on the other hand, are boundary-free, and windowed Fourier transforms provide an excellent framework for approximation and numerical solution of PDEs on these surfaces.

In this talk, a spectral method based on windowed Fourier approximations is presented. It relies on domain decomposition and is suitable for adaptive and parallel implementation. One of the advantages of this approach is that computations can be carried out using fast Fourier transforms on a nearly uniform grid. Approximations are obtained on overlapping domains and a global solution is obtained using the partition of unity. A convergence analysis for analytic functions will be presented.

Minisymposium Organisers

Jie Shen, Purdue University
Mohsen Zayernouri, Michigan State University

Abstract

Rationale: The main objective of this mini-symposium is to present the salient theoretical and applied features of the emerging area of fractional differential equations in addition to new trends in high order methods for anomalous transport. This area of research has received an enormous amount of attention in different areas of mathematics, science, and engineering over the past decade. ICOSAHOM 2018 is a suitable event to organize the proposed mini-symposium at, and it will provide a unique opportunity to invite several influential researchers from all over the world to present recent and cutting-edge advances in this field.

Organization Eight talks will be scheduled for the two 2-hour allotted time periods. The talks will be selected in an attempt to obtain reasonable coverage of all the different theoretical, algorithmic and application areas currently being explored by the spectral/spectral-element, and finite-difference methods community.

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Authors (presenter in bold)

Anna Lischke, Brown University
Guofei Pang, Brown University
Mamikon Gulian, Brown University
Fangying Song, Brown University
Christian Glusa, Sandia National Lab
Xiaoning Zheng, Brown University
Zhiping Mao, Brown University
Wei Cai, Southern Methodist University
Mark M. Meerschaert, Michigan State University
Mark Ainsworth, Brown University
George Karniadakis, Brown University

Abstract

The fractional Laplacian in R^d has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in mathematically distinct ways, and there is currently no consensus in the literature as to which definition of the fractional Laplacian in bounded domains is most appropriate for a given application. The Riesz (or integral) definition, for example, admits a nonlocal boundary condition, where the value of a function u(x) must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. In contrast, the spectral definition requires only the standard local boundary condition. These differences, among others, lead us to ask the question: “What is the fractional Laplacian?” In this presentation, we compare several commonly used definitions of the fractional Laplacian, and we use a quantitative approach to identify their practical differences.

We will provide a quantitative assessment of new numerical methods as well as available state-of-the-art methods for discretizing the fractional Laplacian, and we will present new results on the differences in features, regularity, and boundary behaviors of solutions to equations posed with these different definitions. We will also discuss stochastic interpretations and demonstrate the equivalence between some recent formulations. Through this work, we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications.

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Authors (presenter in bold)

Mohsen Zayernouri, Michigan State University

Abstract

In this talk, we present a new computational-mathematical framework for constructing the fractional-order operators from available data. This is done via formulating an iterative algorithm coupled with an uncertainty quantification framework for capturing the underlying fractional order, model coefficients, also the propagation of their corresponding uncertainties through the fractional model. This computational platform can be readily employed in diverse applications of "anomalous transport".

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Authors (presenter in bold)

Jie Shen, Purdue University (West Lafayette, IN, US)

Abstract

For fractional Laplacian problem in bounded domains, we adopt the Caffarelli-Silvestre extension which transforms the fractional Laplacian equation in d-dimension into an equivalent system with local derivatives in (d+1)-dimension. We develop an efficient numerical method based on the generalized Laguerre approximation in the extended direction and usual (FEM or spectral) approximation in the original domain. Moreover, we enrich the spectral approximation space by using leading singular functions associated with the extended $y$-direction so that high-accuracy can be achieved despite the singularity of extended problem at $y=0$.

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Authors (presenter in bold)

Ke Chen, University of Liverpool

Abstract

Image registration is an useful and yet challenging problem in many applications of Imaging Sciences. In this talk, we present a variational model using regularisers based on fractional derivatives in order to obtain a smooth transformation, competitive to other state of arts models using integer order derivatives. After some analysis, we focus on reformulations to avoid dealing with highly nonlinear and fractional PDEs. The augmented Lagrangian method is employed to derive an effective and iterative algorithm that require the solution of linear PDEs. Advantages and efficiencies of the approach are illustrated with numerical experiments.

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Authors (presenter in bold)

Mohammad Afzal Shadab, Hong Kong University of Science and Technology
Xing Ji, Hong Kong University of Science and Technology
Kun Xu, Hong Kong University of Science and Technology

Abstract

Fifth-order finite volume WENO is proposed for a cylindrical grid which includes derivation of linear weights, optimal weights, and smoothness indicator [1]. The linear and optimal weights are derived using a polynomial approximation in a particular direction (dimension-by-dimension approach) which satisfies the local conservation property. Finally, a Vandermode-like system is obtained, which can be solved for linear weights on both regularly- and irregularly-spaced cylindrical grids, where the analytical relations are derived for the former case. In addition, a simple and straightforward formulation for evaluating the smoothness indicator is derived by minimizing the L2-norm of the derivatives of the reconstruction polynomial and thus, emulating the idea of minimization of the total variation of approximation.

Unlike the Riemann solvers, GKS provides a time-dependent gas-distribution function at the cell interface [2, 3]. By taking conserved moments, time-dependent flux function can be evaluated. This flux function provides a dynamic process of evolution from the kinetic-scale free transport to the hydrodynamic scale wave propagation, which provides the adequate physics for the non-equilibrium numerical shock structure to the near equilibrium Navier-Stokes solution, unlike Riemann solvers, which always employ operator splitting approach [2, 3].

The next big advancement is the two-stage fourth-order (S2O4) technique which significantly reduces the computational cost by reducing the number of intermediate stages requiring expensive spatial reconstructions [3]. Conventional Riemann solvers are first order in time and thus, require the equal number of intermediate stages as of temporal accuracy with distinct spatial reconstructions, e.g., RK order 4 requires four stages. Since the flux is a function of time in GKS, a two-stage Lax-Wendroff type time stepping is proposed recently which achieves fourth- and fifth-order in time with just two stages [3].

In this study, fifth-order finite-volume WENO is developed for cylindrical grids and then applied to Riemann solver-based flux and S2O4 GKS. Several test cases involving one- and two-dimensional smooth and discontinuous flow problems are performed to test the performance of the proposed scheme. This high-order scheme significantly improves the computational cost and quality of the results obtained with GKS over traditional Riemann solvers on cylindrical grids.

[1.]Shadab et.al,arXiv:1711.06212,2017. [2.]Ji et.al,JCP:356:150–173,2018. [3.]Pan et.al,JCP:326:197–221,2016.

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Authors (presenter in bold)

Thien Binh Nguyen, Monash University
Chang-Yeol Jung, Ulsan National Institute of Science and Technology

Abstract

We present a proposed WENO-theta scheme which adaptively switches between a 5th-order upwind and 6th-order central scheme, depending on the smoothness of not only the substencils but also the large one. Unlike the other 6th-order WENO methods in which this switch depends solely on the smoothness of the most downwind sub-stencil, it is that of the large stencil which decides this mechanism in our new scheme. The main features of our new scheme are that the new scheme combines good properties of both 5th-order upwind and 6th-order central schemes. That is, the new scheme is more dispersive than the 5th-order ones in terms of better resolution of small-scale structures and capturing discontinuities. Moreover, our proposed scheme overcomes the loss of resolution around some critical regions for problems which are nonsmooth, that is, the problems whose solution may contain discontinuities or regions with undefined derivatives. Furthermore, the proposed WENO-theta is able to maintain symmetry in the solutions much better than other comparing 6th-order WENO schemes.

We develop a new set of smoothness indicators of the substencils beta_k which are symmetric in terms of Taylor expansions around the point x_j and a new indicator tau for the large stencil. The latter is chosen as the smoother one among two candidates which are computed based on the possible highest-order variations of the reconstruction polynomials in L2 norm. From then, the value of the parameter theta is determined to decide if the scheme is 5th-order upwind or 6th-order central.

A number of numerical tests for both scalar cases, linear and nonlinear, and system case with the Euler equations of gas dynamics is carried out to check the accuracy, resolution, and robustness of our new scheme. This illustrates that the new scheme outperforms other comparing 6th-order WENO schemes in terms of improving the resolution at critical regions of nonsmooth problems as well as maintaining symmetry in the solutions.

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Authors (presenter in bold)

Duarte M. S. Albuquerque, Instituto Superior Técnico, Universidade de Lisboa
Artur G. R. Vasconcelos, Instituto Superior Técnico, Universidade de Lisboa
José C. F. Pereira, Instituto Superior Técnico, Universidade de Lisboa

Abstract

A new very high-order scheme for elliptic operators in unstructured polyhedral grids is proposed, based on the weighted least-squares technique. It can go up to eight-order accuracy and uses face centered reconstructions for the integration of the diffusive fluxes. In order to maintain the local high-order, a new stencil extension algorithm is applied near the boundaries of the computational domain. Additionally, the weight function from the least-squares regressions was optimized according to the matrix condition number, convergence order and error magnitude. From this study, a new weight function is proposed for reconstruction type schemes which achieved the theoretical convergence order for Cartesian, triangular, polyhedral and hybrid grids.

A grid quality study, based on the non-orthogonality angle and volume ratio, proves that the scheme's convergence order is not affected by the grid parameters. To conclude this work two efficiency criteria are used: the required memory and the solver-run time. Both criteria are used to prove the advantages of the eight-order scheme over the other lower order ones. Furthermore, when comparing different grid types, the polyhedral one presented more efficient solutions than its Cartesian and triangular counterparts.

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Authors (presenter in bold)

QIUJU WANG, Institute of Applied Physics and Computational Mathematics
RALF DEITERDING, University of Southampton

Abstract

Spline schemes are proposed to simulate compressible flows on non-uniform structured grid in the framework of finite volume methods. The cubic spline schemes in the present paper can achieve fourth and third order accuracy on the uniform and non-uniform grids respectively. Due to the continuity of cubic spline polynomial function, the inviscid flux can be computed directly from the reconstructed spline polynomial without using the Riemann solvers or other flux splitting techniques. Artificial viscosity models are introduced to damp high frequency numerical disturbances and to enhance the numerical stability. The viscous flux can preserve third order accuracy on both uniform and non-uniform grids with first derivatives directly obtained from the cubic spline polynomials. Then both calculations of the convection and diffusion terms can achieve consistent third-order accuracy on non-uniform grids. A hybrid scheme, in which the spline scheme is blended with shock-capturing WENO scheme, is developed to deal with flow discontinuities. Benchmark test cases of inviscid/viscous flows are presented to demonstrate the accuracy, robust and efficiency of the proposed schemes.

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Authors (presenter in bold)

Christian Klingenberg, Würzburg Universität
Gabriella Puppo, Universita dell'Insubria
Matteo Semplice, Università di Torino

Abstract

In this talk I will focus on novel, arbitrary high-order, finite volume schemes for the Euler equations with a gravitational source term.

The schemes are well-balanced for general hydrostatic equilibria thanks to two key elements. First, the reconstructions of the field variables to generate interface values for evaluating the flux terms, is performed on perturbations of the cell averages from a desired equilibrium, generically described by a pair of functions a(x), b(x) such that the density and pressure of the stationary state are proportional to a(x) and b(x), respectively. In this fashion, when the cell average data being reconstructed represent an equilibrium state, the reconstruction maintains that equilibrium, because the fluctuations are zero, and any polynomial interpolation is able to preserve the null function.

Second, the integration of the source term is carefully chosen in order to match the fluxes. Since the equilibrium prescribes the pressure variable, a reconstruction of the pressure variable is needed. Orders higher than two are achievable only if suitable high order accurate cell averages of the pressure can be computed from the set of cell averages of the conservative variables. In this work, this is achieved by a novel application of the Central WENO (CWENO) reconstruction technique.

The description of the stationary states that is at the core of the well-balancing mechanism encompasses not only the classical isothermal equilibrium, but also the isentropic and more general ones. Both 1D and multi-dimensional cases will be considered in the numerical tests.

Minisymposium Organisers

Stefan Sauter, University of Zurich
Markus Melenk, TU Vienna

Abstract

High-frequency scattering problems have many important applications which include, e.g., radar and sonar detection as well as medical and seismic imaging. Their efficient and reliable numerical modelling is a challenge and the development of fast numerical methods is a topic of vivid worldwide research in numerical analysis and scientific computing. Such problems are often modelled in the frequency domain, where a time-periodic ansatz is employed for the wave or Maxwell equation which, in turn, results in the Helmholtz or stationary Maxwell equation in the high-frequency regime, i.e., with large wave number. High order methods are particularly suited for this problem class. The aim of this minisymposium is to bring together leading experts in the area of the numerical analysis and scientific computing in this area.

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Authors (presenter in bold)

Laurent Demanet, MIT

Abstract

I will review recent progress by many people on the question of solving the Helmholtz equation for propagating high-frequency waves, in linear or sublinear complexity. The question is harder than in the elliptic case, and the better answers seem to involve a decomposition into polarized (one-way) waves. I will describe how the method of polarized traces fits in this framework, and how it can be adapted to lead to a complexity that is sublinear in both the number of volume unknowns and the number of right-hand-sides, in the 3D case, in a parallel environment. High-order HDG variants are also discussed. Joint work with Leo Zepeda, Matthias Taus, Adrien Scheuer, and Russell Hewett.

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Authors (presenter in bold)

Patrick Joly, INRIA
Francis Collino, INRIA
Emile Parolin, INRIA

Abstract

In this work, we investigate new iterative non overlapping domain decomposition for the numerical solution of time harmonic Maxwell's equations. The main novelty of our approach is the use of nonloncal operators (integral operators) in the transmission conditions between two adjacent subdomains, in order to ensure an exponential rate of convergence, in the spirit of previous works for the scalar Helmholtz equation. The main theoretical aspects of the method will be detailed and preliminary numerical results will be presented.

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Authors (presenter in bold)

Markus Melenk, TU Vienna
Stefan Sauter, University of Zurich

Abstract

We consider the time-harmonic Maxwell equation in homogeneous media at large wavenumbers $k$ discretized by high order edge elements. For a model problem, we show that quasi-optimality is achieved if (a) the approximation order $p$ is selected as $p = O(\log k)$ in conjunction with (b) a mesh size $h$ such that $kh/p$ is small. As in the related case of the Helmholtz equation, the analysis relies on a $k$-explicit regularity theory for Maxwell's equations that decomposes solutions into two components: the first component is an analytic, but highly oscillatory function while the second one has finite regularity but features wavenumber-independent bounds. An issue particular to Maxwell's equations that is not present in the case of the Helmholtz equations is the problem of approximating discrete divergence-free functions by divergence-free ones. In the present $hp$-version context, this is possible at the optimal rate due to a suitable projection-based interpolation operator that enjoys the commuting diagram property.

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Authors (presenter in bold)

Francesca Rapetti, Universite de la Cote Azur

Abstract

We follow the geometric point of view detailed in [RB09] to define the generators of the well known H(curl)- and H(div)-conforming Nedelec first type spaces of degree r > 1. The focus here is on one mesh simplex but the fact of relying on barycentric coordinates makes the local construction independent (up to a re-orientation step) of the considered mesh element. With the list of generators and an unisolvent set of degrees of freedom, we recall how it is possible to compute a cardinal basis for these mixed finite element spaces by relying on the inversion of a generalized Vandermonde matrix [BzR16]. We finally focus on edge finite elements to illustrate the numerical features (accuracy, conditioning, dispersion) of this methodology when applied to a problem of mode propagation in a wave-guide.

References:

[BzR16]. M. Bonazzoli and F. Rapetti, High order finite elements in numerical electromagnetism: degrees of freedom and generators in duality Numerical Algorithms, Vol. 74, No. 1, p. 111-136, 2017

[RB09] F. Rapetti and A. Bossavit, Whitney forms of higher degree SIAM J. on Numerical Analysis, Vol. 47, No. 3, pp. 2369-2386, 2009.

Minisymposium Organisers

Jae-Hun Jung, SUNY Buffalo, Ajou University
Sigal Gottlieb, UMass Dartmouth

Abstract

Data analysis such as handling, interpretation, and utilization of data emerges in recent years as a crucial component in most research areas of science and humanity. The proposed mini-symposium attempts to see how research in high order methods could play a role in the era of data science. This mini-symposium will bring together researchers in the high order research community whose current research finds data analysis a critical part of their research. It will also provide a unique opportunity for researchers in high order methods area to find a possible research agenda in data science by sharing their expertise and experiences. In this mini-symposium recent progress of data analysis methods in dealing with data created by high order methods will be introduced. The mini-symposium will also present recent works of deep learning, topological data analysis of spectral approximations and their applications.

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Authors (presenter in bold)

Akil Narayan, University of Utah

Abstract

Numerical quadrature rules that use point values are ubiquitous tools for approximating integrals. Some of the most popular rules achieve accuracy by enforcing exactness for integrands in a finite-dimensional polynomial space. When the integration domain is one-dimensional, classical rules are available and plentiful. In multidimensional domains with non-standard polynomial spaces and weights, the situation is far more complicated. We will present a general methodology for numerically generating approximate polynomial quadrature rules in multidimensional case that can achieve high-order accuracy when used to solve scientific problems. Our strategy for design of quadrature rules utilizes several tools from the emerging field of data science: restricted invertibility, subset selection, probabilistic concentration estimates, sampling design in high dimensions, and nonconvex optimization.

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Authors (presenter in bold)

Jae-Hun Jung, SUNY Buffalo, Ajou University
John Nicponski, SUNY Buffalo

Abstract

Vascular disease is the leading cause of death world-wide. About 17.3 million people die every year and the number of deaths is expected to exceed 23.6 million by 2030. Better understanding and classifying of the types of vasculature can help save more lives. Diagnosis is based on the anatomical approach using the angiography and/or direct measurement of the fractional flow reserve with catheter or computational fluid dynamics. Numerical indices obtained by these methods, however, do not necessarily provide clear interpretation of the vasculature unless the case is extreme and the intervention is inevitable.

In this talk, we present a new additional approach using the topological data analysis (TDA) for the stenotic vascular flows. The key element of the topological data analysis for vascular disease presented in this talk is to project the physical data unto the n-dimensional unit sphere and then find the number of generators of the homology group at each dimension. For the vascular data, we use the spectral method and solve the 3D incompressible Navier-Stokes equations. The obtained data is projected onto the n-sphere and we apply the TDA to the projected data. That is, we calculate the Betti number in each dimension of the projected spectral approximation data and measure the "persistence" as a numeric index of the degree of disease. In addition to homology group, we also propose to use the generalized percent stenosis by measuring the persistence in 0-dimension. The 3D barcode is introduced to check the reliability of the persistence.

By generating more refined numerical indicators, the proposed method could provide more definite interpretation in the ambiguous range of the numerical indices and more accurate diagnosis of the disease.

Minisymposium Organisers

Misun Min, Argonne National Laboratory
Paul Fischer, University of Illinois, Argonne National Laboratory
Tzanio Kolev, Lawrence Livermore National Laboratory

Abstract

High-order discretizations have the potential to provide an optimal strategy for achieving high performance and delivering fast, efficient, and accurate simulations on next-generation architectures. Traditional low-order schemes, developed in an era when FLOPs were the performance-limiting factor, will no longer be an attractive option, since the cost of FLOPs will be considerably smaller than the cost of data motion. While many high-order methods were intractable on older architectures, their importance and usefulness has been dramatically increasing and has demonstrated significant improvements in both simulation quality and parallel strong scalability.

This session will discuss the next-generation high-order discretization algorithm and software, based on finite/spectral element approaches that will enable a wide range of important scientific applications to run efficiently on future architecture.

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Authors (presenter in bold)

Florian Hindenlang, Max-Planck Institute for Plasma Physics
Eric Sonnendrücker, Max-Planck Institute for Plasma Physics
Gregor Gassner, University of Cologne
Andrew Winters, University of Cologne

Abstract

The class of high order Discontinuous Galerkin (DG) methods is a very promising candidate for large-scale simulations of nonlinear advection-diffusion type problems, found for example in fluid dynamics or magneto-hydrodynamics. DG methods combine high order accuracy with a flexible discretization using unstructured meshes. Elements couple only to direct face neighbors, via numerical flux functions. When explicit time stepping is used, the locality of the high order operator makes the algorithms well suited for large scale parallel simulations.

The discontinuous Galerkin spectral element method (DGSEM) is an interesting DG variant for three-dimensional simulations on curvilinear meshes. DGSEM collocates interpolation and integration points (Gauss or Gauss-Lobatto nodes) on tensor-product hexahedral elements. The resulting operator keeps the tensor-product structure and therefore significantly reduces the computational cost of element-local evaluations for high polynomial degrees, so that the measured computational time per degree of freedom per time update becomes nearly independent of the polynomial degree.

However, aliasing errors will occur when applying collocation methods to nonlinear problems. Together with the low dissipation and dispersion errors of a high order scheme, the aliasing strongly affects the robustness of the method, which can even lead to crashing of a simulation for low resolutions.

More recently, discrete stability proofs from Finite Difference summation-by-parts operators have been extended to DGSEM with Gauss-Lobatto nodes, where the use of so-called two-point fluxes in the volume and surface integrals lead to a significantly increased robustness. The two-point fluxes can be designed to discretely preserve kinetic energy or discretely fulfill the thermodynamic entropy inequality, but are more costly to evaluate.

In this talk, we present the open-source code FLUXO (github.com/project-fluxo/fluxo), jointly developed by F. Hindenlang at IPP Garching, the groups of G. Gassner at University of Cologne and of C-D. Munz at the University of Stuttgart. We will discuss the efficient implementation of the stable two-point flux approach for simulations with curved three-dimensional elements. In addition, we give details about the parallelization with a non-blocking communication strategy to hide communication latency by local computations. Finally, the application of FLUXO for large-scale 3D MHD simulations of a torus-shaped strongly magnetized plasma will be shown.

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Authors (presenter in bold)

Chris Cantwell, Imperial College London
David Moxey, University of Exeter
Michael Turner, Imperial College London
Martin Vymazal, Imperial College London
Douglas Serson, Universidade de São Paulo (USP)
Joaquim Peiro, Imperial College London
Mike Kirby, University of Utah
Spencer Sherwin, Imperial College London

Abstract

High-order spectral/hp element methods are emerging as a potential technology for achieving unprecedented levels of detail for industrial simulations, particularly in the aeronautical and automotive sectors. The lower dissipative nature of these methods allow complex transient flow features, such as near-wall vortices, to be more accurately captured and propagated over longer length-scales than lower-order methods. These algorithms also present favourable numerical properties, such as high arithmetic intensity, which make them align well with massively parallel computational resources and modern system architectures.

In this talk, we will present the latest and ongoing efforts of the Nektar++ team to address the challenges which need to be overcome in order to enable these scale-resolving simulations to be undertaken successfully, thereby bringing these methods closer to industrial readiness. These will be illustrated with some of our recent large-scale runs.

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Authors (presenter in bold)

Mark Taylor, Sandia National Labs
David Hall, UC Boulder
Andrew Steyer, Sandia National Labs
Paul Ullrich, UC Davis
Oksana Guba, Sandia National Labs

Abstract

We will describe a spectral element non-hydrostatic dynamical core being developed for the Energy Exascale Earth System Model (E3SM). One of the key goals of the E3SM project is to develop a coupled climate model that can make efficient use of upcoming Exascale computing platforms. Spectral elements remain an attractive approach for these new architectures. Advantages include a compact stencil with high data reuse, a tensor product decomposition which avoids indirect addressing even with highly unstructured meshes, a high work to communication ratio, and all communication isolated to a single kernel minimizing the amount of code optimization needed.

Our atmospheric dynamical core is based on the HOMME hydrostatic dynamical core. It shares many of the underlying computational kernels including full support for variable resolution meshes. We use the Laprise mass coordinate formulation of the equations with the shallow atmosphere approximation. This allows us to support both hydrostatic and non-hydrostatic formulations as a run time option. The nonhydrostatic formulation adds two additional prognostic variables (vertical velocity and geopotential height). The discretization relies on mimetic methods: spectral finite elements in the horizontal, finite differences with a Lorenz staggering in the vertical, and both vertically Lagrangian and Eulerian options. We use a horizontally-explicit vertically-implicit (HEVI) IMEX approach for the time discretization. We introduce a moist potential temperature as a prognostic variable, and formulate the equations so that total energy are conserved through the use of mimetic discretizations. Initial results will be presented from several of the DCMIP idealized test cases. Initial performance results will be presented from the Intel KNL supercomputer at NERSC.

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Authors (presenter in bold)

Frank Giraldo, Naval Postgraduate School
Michal Kopera, University of California at Santa Cruz
Jeremy Kozdon, Naval Postgraduate School

Abstract

We present a unified continuous and discontinuous Galerkin method for constructing atmospheric and ocean nonhydrostatic models. Results for our atmospheric component (called NUMA) have been presented and published previously. However, our work on the ocean component (called NUMO) is relatively new. We will describe our attempts to construct adaptive mesh refinement into both the NUMA and NUMO models with a particular emphasis on test cases to verify the models.

Minisymposium Organisers

Bryan Quaife, Florida State University
Travis Askham, University of Washington
Ludvig af Klinteberg, Simon Fraser University

Abstract

In the modern computing environment, high-order methods are of increasing importance for the accurate simulation of complex fluid flows. Applications include porous media flow, suspensions of rigid and deformable particles, and multiphase flows. Numerical simulations of such complex systems are challenging because of strong nonlinearities, non-local interactions, time-dependent geometries, multiple length and time scales, and fluid-structure interactions. Integral equation methods offer a powerful and flexible framework that addresses many of these challenges while providing high-order accuracy. This symposium will discuss recent advances in high-fidelity and scalable integral equation methods in fluid dynamics.

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Authors (presenter in bold)

Ludvig af Klinteberg, Simon Fraser University

Abstract

Integral equation methods provide an efficient way of solving elliptic PDEs to high accuracy, particularly in complex geometries. They are however limited to homogeneous problems, and have in the context of fluid dynamics mainly been used for solving the Stokes equations. Recent years have seen development in the use of integral equation methods also for inhomogeneous problems. In these hybrid methods, the solution is represented as a sum of a layer potential and a volume potential.

We will in this presentation discuss a high-order, hybrid integral equation method for the incompressible Navier-Stokes equations in 2D. The method is based on fast algorithms for layer potentials and volume potentials, and is suitable for problems with complex geometries and moderate Reynolds numbers.

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Authors (presenter in bold)

Chiara Sorgentone, KTH Royal Institute of Technology
Anna-Karin Tornberg, KTH Royal Institute of Technology

Abstract

We consider 3D drops covered by an insoluble surfactant and immersed in a liquid with different viscosity and subjected to external flow and electric fields. Surfactants (surface active agents) are compounds that lower the surface tension between liquids; they are widely used in engineering applications, in pharmaceuticals, foods and petroleum industries. There is a growing interest also in the applications of electric-field induced dynamics on deformable particulate suspensions. Biomedical applications span from separation and detection (for example of infected blood cells, DNA and protein molecules), to selective manipulation, drug delivery and so on. Other engineering applications are represented by mixed emulsions where a specific material needs to be isolated. It is then clear the increasing need of designing biotechnological devices for such purposes and how fundamental is to understand the physics of these systems.

The dynamic of the drops is governed by the Stokes equations coupled with the electric potential equation and the surface convective-diffusion equation for the evolution of the surfactant concentration. With the large surface to volume ratio of these drops, the dynamics at the interface become extremely important and it is very natural to consider integral equations for this problem.

The proposed method is able to simulate 3D drops with different viscosities and close interactions. An accurate representation of both the drop surfaces and the surfactant concentration is given by a spherical harmonics expansion and a new reparameterization method is introduced to maintain a high-quality representation of the drops also under deformation. Specialized quadrature methods for singular and nearly singular integrals that appear in the formulation are evoked and the adaptive time stepping scheme for the coupled drop and surfactant evolution is designed with special attention to the implicit treatment of the surfactant diffusion.

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Authors (presenter in bold)

Dhairya Malhotra, New York University
Antoine Cerfon, New York University
Mike O'Neil, New York University

Abstract

We consider the problem of computing ideal magneto-hydrodynamic (MHD) equilibrium in intrinsically three-dimensional magnetically confined plasmas. Specifically, we focus on problems with stepped pressure profiles for which the equilibrium problem is known to be well-posed. We partition the problem into regions of constant pressure separated by flux surfaces and the solution in each region is given by a Taylor relaxed state. The equilibrium position of the flux surfaces in this formulation is not known and must be solved to satisfy the pressure balance equation at each point on the surfaces.

In our scheme, we compute Taylor states using the generalized Debye source formulation for the time-harmonic Maxwell's equations, which results in a well-conditioned second-kind boundary integral equations. The boundary integral formulation requires fewer unknowns since we do not need to discretize the entire volume and this results in significant savings in work. However, the computation of the boundary integral operator requires efficient high-order quadratures for singular and near singular integrals on complex three-dimensional surfaces. We will discuss efficient parallel algorithms for such quadratures. We use pseudo-transient continuation to solve the non-linear pressure balance equation and determine the position of the flux surfaces. To ensure that the surfaces are accurately resolved at all times during the non-linear solve for the we reparameterize the surface meshes.

The overarching goal of this work is to analyze the equilibrium and stability properties of plasmas in stellarators and to improve existing designs to achieve better plasma confinement while also simplifying the shape of coils to lower construction costs.

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Authors (presenter in bold)

Timo Betcke, University College London

Abstract

High-fidelity solutions of ever more complex boundary element problems make necessary the optimal use of heterogeneous computing devices. In this talk we present ongoing work on Bempp-cl, an open-source boundary element framework, completely based on OpenCL, which is designed to scale well on modern SIMD based CPUs, and on many-core GPU accelerators. We discuss choice of integration rules, data-layouts, and workgroup setup, and present several benchmark examples on CPU systems, and AMD and Nvidia GPU devices.

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Authors (presenter in bold)

Juan Manzanero, ETSIAE-UPM (School of Aeronautics in Madrid)
Esteban Ferrer, ETSIAE-UPM (School of Aeronautics in Madrid)
Gonzalo Rubio, ETSIAE-UPM (School of Aeronautics in Madrid)
Eusebio Valero, ETSIAE-UPM (School of Aeronautics in Madrid)

Abstract

We analyse the numerical dissipation introduced by three strategies typically considered for implicit LES in high order discontinuous Galerkin methods: upwind Riemann solvers, viscous discretizations for second order terms (BR1 and Interior Penalty), and spectral vanishing viscosity (SVV). First, we analyze these methods using a linear von Neumann analysis (for linear advection equations) to characterize their properties in wavenumber space. Second, we validate these observations on the 3D Taylor-Green vortex (TGV) Navier-Stokes problem with transitional/turbulent flow.

We show that the dissipation introduced by upwind Riemann solvers affects high wavenumbers and that it is neither linear nor monotonic. This dissipation may be increased until a critical high value to then decrease as the discretization tends to that of a conforming (i.e. continuous Galerkin) method.

Regarding the dissipation introduced by the discretisation of elliptic terms (second order derivatives), we show that this dissipation acts at low and medium wavenumbers (lower to the region affected by the Riemman solvers).

Finally, we study the dissipation introduced by SVV operators on discontinuous Galerkin methods. We found that with an appropriate kernel (that selects the modes where artificial viscosity is applied) it is possible to control the dissipation introduced in the low and medium wavenumber region.

Combining these ideas, we propose a new SVV approach that uses a Smagorinsky based viscosity to compute the appropriate numerical viscosity (which varies with the wavenumber). When this viscous kernel is combined with a low dissipation Riemann solver, the method is capable of maintaining low dissipation levels in laminar flows, whilst modelling small eddies and providing the correct dissipation at all wavenumber ranges.

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Authors (presenter in bold)

Clinton P. T. Groth, University of Toronto
Lucie Freret, University of Toronto
Ral Bielawski, University of Toronto

Abstract

The application of a novel high-order, central essentially non-oscillatory (CENO), finite-volume scheme with block-based anisotropic adaptive mesh refinement (AMR) to large-eddy simulation (LES) of turbulent premixed flames is considered. The block-based AMR and high-order CENO methods are applied to the solution of the Favre-filtered Navier-Stokes equations governing a fully-compressible reactive mixture on multi-block, body-fitted, computational mesh consisting of hexahedral elements. The block-based AMR scheme allows enhancement of the local mesh resolution, with preference given to directions as dictated by the flow solution. The CENO method uses a hybrid reconstruction approach based on a fixed central stencil that avoids the complexities of other ENO schemes and offers several computational advantages. The proposed methodology is first validated for a laboratory-scale Bunsen-type premixed burner with methane as the fuel and then numerical results are discussed for the prediction of the bluff-body-stabilized lean premixed propane-air flame. The performances of the combined AMR and high-order approach relative to non-adapted meshes and a standard lower-order scheme are assessed and discussed.

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Authors (presenter in bold)

Miguel A. Moratilla-Vega, Loughborough University
Vishal Saini, Loughborough University
Hao Xia, Loughborough University
Gary J. Page, Loughborough University

Abstract

Jet noise has been one of the most important areas of research since the early days of commercial aviation due to its large contribution to the overall noise generated by an aircraft. During the last decades, the increase of computational power and the development of better numerical methods have made possible the use of CFD simulations as an accurate and efficient tool for its study. Traditionally, CFD combined with surface integral methods has been the preferred procedure to study jet noise because of its reduced cost. However, these methods lack insight of the sound generation, which is a fundamental aspect to understand and reduce the noise. This study uses a different two- steps approach that combines a large-eddy simulation (LES) to obtain the acoustic sources from the flow field, and an acoustic perturbation equations (APE) code that propagates the sound from the near to the far field. A coupling mechanism has been implemented so that the sources coming from the flow simulations are interpolated into the acoustic domain.

The LES is performed with a density-based, second-order finite volume solver that employs the sigma-subgrid-scale model. The APE code uses a high-order Discontinuous Galerkin spectral/hp finite element solver which is part of the Nektar++ framework. By using a high- order wave propagation code low numerical dissipation, hence minimum damping of the waves, is achieved. The APE does not suffer from instability issues related to the more common Linearised Euler Equations (LEE). Moreover, the APE is advantageous by means of a filtered source to eliminate pseudo-sound. The code has non-reflecting boundary conditions combined with a sponge zone that avoids the generation of spurious noise reflected back into the domain.

The proposed work uses a novel communication tool to couple the two codes. This ensures that the resources required for the overall process are optimised. In addition, a comparison between different expansion types and polynomial orders has been carried out to estimate the most efficient set of parameters for jet noise propagation when using a spectral/hp DG method. Additional results will be shown for high Reynolds number jets in isolated and installed configurations.

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Authors (presenter in bold)

Filipe Fabian Buscariolo, Imperial College, USP
Spencer Sherwin, Imperial College
Gustavo Assi, USP
Julio Meneghini, USP
Denis Doorly, Imperial College

Abstract

The Ahmed Body is one of the most widely studied bluff bodies used for automotive conceptual studies and Computational Fluid Dynamics (CFD) software validation. This work focuses on the correlation study between a computational and physical model of an Ahmed Body with slant angle of 25 degrees, which generates a 3D flow on the back, based on a wind tunnel with moving ground and Reynolds number of 1.7M, based on the length of the body. The cases studies three high-order meshes: 4th, 5th and 6th; combined with three polynomial orders: 4th, 5th and 6th for a base mesh setup with around 90,000 elements and another mesh (h) refined setup with around 250,000 elements, summarising a total of 18 cases, considering a SVV-LES model in Nektar++. In order to improve the computational costs, only half of the model is simulated and a symmetry condition is imposed on the centreline. Considering the drag coefficient values for converged cases on the 5th and 6th polynomial order, the maximum difference found, compared with experimental results with moving ground configuration, there is a maximum difference of 16%, however the simulation does not take into account the upper support used in the experimental setup. Comparing Spectral/hp element SVV-LES case from literature, the agreement with the experimental drag coefficient has been improved, reducing the gap from 45% to 16%. For the lift coefficient the maximum difference between the simulation results compared to experimental data is 3%. There is also a good agreement between the LDA measurements on the end of the body with the results from the simulation. It is possible to observe a more intense vortex core on the simulation results, as compared to experimental data, which might well be explained by the upper support used to fix the Ahmed body in test, which weakens the vortices. The methodology shows promising results against the open literature, once an appropriate validation study has been undertaken. Despite the relatively course resolution adopted the results are encouraging. Having identified an appropriate resolution, we will next consider other slant angles, to see how well these correlate with the experimental studies.

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Authors (presenter in bold)

Viktor Linders, Technion - Israel Institute of Technology
Yann T. Delorme, Technion - Israel Institute of Technology
Kunal Puri, Technion - Israel Institute of Technology
Steven H. Frankel, Technion - Israel Institute of Technology

Abstract

Numerical methods for partial differential equations satisfying the summation-by-parts (SBP) property has received considerable attention due to provable energy stability, if boundary conditions are imposed in a weak sense. Recently, the framework has also been used to demonstrate entropy conservation for non-linear problems (Fisher et al. 2013). While originally developed for finite difference discretisations (Kreiss, Scherer 1974) the SBP framework has been extended to spectral methods (Carpenter, Gottlieb 1996), finite volume methods (Nordström, Björck 2001), discontinuous Galerkin spectral elements (Gassner 2013) and flux reconstruction (Ranocha et al. 2016).

Cast in the SBP framework, the above methods yield similar energy estimates, suggesting a close connection among their behaviours in different numerical settings. Hence, it is natural to ask to what extent the behaviour of these schemes can be attributed to the utilisation of the common SBP framework, in contrast to the intrinsic differences among the methods. It is therefore of interest to compare and contrast the performance of some of these methods for relevant test problems.

In this contribution we compare the performance of high order SBP finite difference (FD) methods and flux reconstruction (FR) in the context of an under-resolved turbulence model. As our test problem we consider the Taylor-Green vortex (TGV), governed by the fully compressible Navier-Stokes equations. The relatively simple TGV flow case illustrates the transition to turbulence through the production of small eddies and the enhancement of kinetic energy dissipation. We study the development of the kinetic energy content and dissipation rate for the FD and FR schemes in an under-resolved setting as compared to reference data in the form of a high resolution direct numerical solver (van Rees et al. 2011). The study grants insights into the performance similarities and differences between the FD and FR schemes when used for implicit large eddy simulations.

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Authors (presenter in bold)

Andrea Cassinelli, Imperial College London
Francesco Montomoli, Imperial College London
Paolo Adami, Rolls-Royce
Spencer Sherwin, Imperial College London

Abstract

The high order spectral/hp element methods implemented in the software framework Nektar++ are investigated for scale-resolving simulations of LPT profiles. There is a growing demand for high fidelity methods for turbomachinery to move towards numerical "experiments". The study contributes at building best practices for the use of emerging high fidelity spectral element methods in turbomachinery predictions, with focus on the numerical details that are specific of these classes of methods. For this reason, the T106A cascade is used as a base reference application because of availability of data and results from previous investigations. The effects of polynomial order (p-refinement), spanwise domain extent and spanwise Fourier planes are considered, looking at flow statistics, convergence and sensitivity of the results. The performance of the high order spectral/hp element method is also assessed through validation against experimental data at moderately high Reynolds number. Thanks to the reduced computational cost, the proposed methods will have a strong impact in turbomachinery, paving the way to its use for design purposes and also allowing for a deeper understanding of the flow physics.

Minisymposium Organisers

Yulong Xing, Ohio State University
Xiangxiong Zhang, Purdue University

Abstract

The focus of this special session will be on recent developments in the design, analysis, implementation and application of structure preserving numerical methods, such as positivity/bound preserving, energy preserving, vorticity preserving, entropy satisfying, well-balanced and asymptotic preserving methods. The special session aims to bring together researchers from various fields to report the recent developments in structure preserving methods and to discuss the relevant applications. The topics will span a range from theoretical results to novel algorithms, and will include talks focused on a variety of interesting application areas.

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Authors (presenter in bold)

Alexander Kurganov, Southern University of Science and Technology

Abstract

We are interested in deriving robust asymptotic preserving (AP) numerical methods for Euler equations of gas dynamics and shallow water equations with Coriolis forces. It is well-known that both the Euler and shallow water equations become stiff in low Mach and low Froude number regimes, respectively. In these regimes, the applicability of explicit schemes is limited due to severe stability limitation on the mesh size: $\Delta x \sim \varepsilon$ and $\Delta t \sim \varepsilon^2$, where $\varepsilon$ is the Mach/Froude number. In order to design substantially more efficient numerical methods, one typically needs to design an AP scheme, which approximates the incompressible equation obtained in the limiting $\varepsilon \to 0$ case. Such schemes can be designed using either a flux splitting implicit-explicit (IMEX) or fully implicit approach. In this talk, I will discuss both approaches and present several different AP schemes.

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Authors (presenter in bold)

Christian Klingenberg, Wuerzburg University

Abstract

Finite volume methods for compressible Euler equations suffer from an excessive diffusion in the limit of low Mach numbers. This lecture explores approaches to overcome this.

We begin with the the acoustic equations obtained as a linearization of the Euler equations. The limit of compressible to incompressible flows is characterized by a divergence-free velocity. Also in multiple space dimensions advection and acoustics are genuinely different. These concepts are found to have a counterpart in the discrete setting and to be at the origin of the difficulties at resolving the low Mach number limit. We call numerical schemes whose discrete stationary states discretize all the analytic stationary states of the PDE 'stationarity preserving'. It is shown that for the acoustic equations, stationarity preserving schemes are also vorticity preserving and are also asymptotic preserving for the Mach number going to zero. This establishes a new link between these three concepts. We identify all those stencils that are discretizations of the divergence that allow for stabilizing stationarity preserving diffusion. This way we are able to characterize all stable discretization of the acoustic equation that are asymptotic preserving for low Mach numbers.

Stationarity preservation can be generalized to the nonlinear Euler equations. We present numerical schemes for the Euler equations that are stationarity preserving. This leads to a low Mach number asymptotic preserving numerical scheme. Its diffusion is chosen such that it depends on the velocity divergence rather than the derivatives of the different velocity components. This is demonstrated to overcome the low Mach number problem. The scheme shows satisfactory results in numerical simulations and it has been found to be stable under explicit time integration.

This is joint work with Wasilij Barsukow.

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Authors (presenter in bold)

Alina Chertock, North Carolina State University

Abstract

In this talk, I will discuss recent advances in the development of structure preserving numerical methods for nonlinear hyperbolic systems of conservation and balance laws. In particular, I will present new well-balanced and positivity preserving numerical schemes, that is, the methods which are capable of exactly preserving some steady-state solutions as well as maintaining the positivity of the numerical quantities when it is required by the physical application. The new schemes are based on modifications in the reconstruction and evolution steps of Godunov-type central-upwind methods. A crucial step of this modification is to introduce an equilibrium variable obtained from incorporating the source term into the flux and then evolving them in time according to a well-balanced central-upwind evolution in time, which is adapted to reduce the amount of numerical viscosity when the flow is at (near) steady-state regime. I will demonstrate the performance of our newly developed central-upwind schemes and demonstrate importance of perfect balance between the fluxes and gravitational forces in a series of one- and two-dimensional examples.

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Authors (presenter in bold)

Jean-Luc Guermond, Texas A&M
Bojan Popov, Texas A&M
Ignacio Tomas, Texas A&M

Abstract

A second-order finite-element-based method for approximating the compressible Euler equations is introduced. The method preserves all the known invariant domains of the Euler system: positivity of the density, positivity of the internal energy and the local minimum principle on the specific entropy. The technique combines a first-order, invariant domain preserving, Guaranteed Maximum Speed method using a Graph Viscosity (GMS-GV1) with an invariant domain violating, but entropy consistent, high-order method. Invariant domain preserving auxiliary states, naturally produced by the GMS-GV1 method, are used to define local bounds for the high-order method which is then made invariant domain preserving via a convex limiting process. Numerical tests confirm the second-order accuracy of the new GMS-GV2 method in the maximum norm, where 2 stands for second-order. The proposed convex limiting is generic and can be applied to other approximation techniques and other hyperbolic systems.

Minisymposium Organisers

Gustaaf Jacobs, San Diego State University
Wai-Sun Don, School of Mathematical Sciences, Ocean University of China
David Kopriva, San Diego State University

Abstract

High order linear methods — such as spectral methods, optimized compact schemes, optimized central finite difference schemes, and discontinuous Galerkin methods — are widely used for solving PDEs with smooth solutions because of their efficient resolution and computation of both large and fine scale structures, including those existing in turbulence. However, the methods are usually applied, in practice, when the flow is marginally resolved or in the presence of shocks or high gradients that arise from the well-known Gibbs phenomenon. In contrast, schemes that regularize and stabilize — such as the weighted essentially non-oscillatory scheme (WENO), Discontinuous Galerkin with limiters and Spectral Domain Decomposition with physical based artificial viscosity— resolve discontinuities and high gradients in an essentially non-oscillatory manner, while also computing smooth structures with high resolution. In this minisymposium, we gather experts from the community to share their recent research and views with the goals of advancing high order methods to solve PDEs in marginal regimes and to discuss applications these methods can reliably compute.

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Authors (presenter in bold)

Chi-Wang Shu, Brown University

Abstract

Positivity-preserving discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws have been extensively studied in the last several years. But nearly all the developed schemes are coupled with explicit time discretizations. Explicit discretizations suffer from the constraint for the Courant-Friedrichs-Levis (CFL) number. This makes explicit methods impractical for problems involving unstructured and extremely varying meshes or long-time simulations. Instead, implicit DG schemes are often popular in practice, especially in the computational fluid dynamics (CFD) community. In this talk we discuss a high-order positivity-preserving DG method with the backward Euler time discretization for conservation laws. Both the analysis for the simple linear case and numerical experiments for nonlinear systems indicate that a lower bound for the CFL number is required to obtain the positivity-preserving property. The proposed scheme not only preserves the positivity of the numerical approximation without compromising the designed high-order accuracy, but also helps accelerate the convergence towards the steady-state solution and add robustness to the nonlinear solver. This is a joint work with Tong Qin.

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Authors (presenter in bold)

Wai Sun Don, Ocean University of China

Abstract

Ever since the development of the high order WENO finite difference scheme by Jiang and Shu, it has been widely adopted for solving hyperbolic conservation laws that captures shocks essentially non-oscillatory while resolving fine scale structures efficiently. Numerous advances have been proposed to reduce the dissipation and dispersion errors by improving/modifying certain critical components of the WENO reconstruction procedure, such as, the definitions of nonlinear weights and the smoothness indicators and the tuning of existing parameters. In this talk, I will present some recent topics and results regarding the positivity-preserving property of the 5th and 7th order WENO FD scheme with global, local Lax-Fredrichs, and Roe/HLLC flux splitting for some often cited benchmark problems under extreme conditions. Furthermore, despite the enhancement of the scheme, hybrid scheme that conjugates the high order linear compact and nonlinear WENO FD scheme can be employed to improve the performance of the scheme in the smooth regions of the solution. I will introduce several key components and enhancements to improve the robustness of the high order shock detection algorithms based on, the high order polynomial based multi-resolution analysis, the trigonometric conjugate Fourier analysis, and the Radial basis function based edge detection algorithm. Classical examples in shallow water equation with shock-like hydraulic jumps based on the well-balanced hybrid schemes and Euler equations for compressible gas dynamics and detonation wave simulations based on the bound preserving hybrid schemes will be presented to demonstrate the performance of the hybrid schemes in terms of resolution power, dissipation, and efficiency.

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Authors (presenter in bold)

Bart Wissink, San Diego State University
Gustaaf Jacobs, San Diego State University
Jennifer Ryan, University of East Anglia
Wai-Sun Don, Ocean University of China
Edwin van der Weide, University of Twente

Abstract

We present a shock capturing method that is based on a smoothness-increasing and accuracy-preserving (SIAC) filter. A SIAC filter in polynomial form is developed for regularization of discontinuities in general higher-order bases. We focus on a Chebyshev basis and approximation of hyperbolic equations on a single domain and in collocation from.

Through moment conditions, the filter ensures a formal convergence away from discontinuities. The convergence rate can be proven to be according to the number of zero moments plus one. If the projection on the underlying basis is exact then the compact support can be chosen arbitrary small to satisfy the moment conditions. The only heuristic requirement is that the filter is supported by at least three spectral collocation points.

The filter implements algebraically and combines with the collocation differentiation matrix in a pre-processing stage and hence does not incur computational expense on the time-dependent solution of the hyperbolic equation.

For linear advection equations, we verify that the filter gives an order convergence according the number of zero moment of the filter. For non-linear equations including the Burger's equation and the Euler equations in one and two-dimensions, higher resolution is obtained.

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Authors (presenter in bold)

sebastiano Boscarino, university of Catania
Jingmei Qiu, university of delaware
giovanni russo, university of Catania
tao xiong, xiamen university

Abstract

We propose a high order WENO method coupled with IMEX time discretization for all-Mach simulations. A proper implicit treatment on the stiff pressure term allows for a large time stepping size with stability, compared with the regular CFL time step restriction for explicit schemes. Further more, such design of the scheme ensures the correct asymptotic limit when the Mach number goes to zero, leading to a consistent and high order scheme for limiting incompressible Euler system. 1D and 2D numerical results will be presented to showcase the proposed schemes and their properties.

Minisymposium Organisers

Henar Herrero, Universidad de Castilla-La Mancha
Juan Sánchez-Umbría, Universitat Politècnica de Catalunya

Abstract

The main objective of this minisymposium is to present some recent advances in high-order and spectral methods applied to hydrodynamic stability problems, where these methods play a relevant role due to their capability to produce very accurate numerical results. These problems include solving directly the Navier-Stokes equations, stability and eigenvalue problems, continuation of bifurcations techniques, etc.

Different numerical tecniques, computing implementations and applicationes will be presented, such as reduced basis, preconditioning, continuation, Rayleigh-Bénard, Taylor-Couette, bubble instabilities, ...

This minisymposium will bring together scientists to present their recent advances on algorithm development and analysis of high-order and spectral methods applied to hydrodynamic instability problems, provide a forum for discussion and interaction in these methods.

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Authors (presenter in bold)

Laurette Tuckerman, CNRS
Jacob Langham, University of Bristol
Ashley Willis, University of Sheffield

Abstract

Steady states and traveling waves play a fundamental role in understanding hydrodynamic problems. Even when unstable, these states provide the bifurcation-theoretic explanation for the origin of the observed states. In turbulent wall-bounded shear flows, these states have been hypothesized to be nodes organizing the trajectories within a chaotic attractor. Numerical computation of these flows must performed with Newton's method, or one of its generalizations, since time-integration cannot converge to unstable equilibria. The bottleneck is the solution of linear systems involving the Jacobian of the Navier-Stokes or Boussinesq equations. Originally such computations were carried out by constructing and directly inverting the Jacobian, but this is unfeasible for the matrices arising from three-dimensional hydrodynamic configurations in large domains. A popular method is to seek states that are invariant under numerical time integration. Surprisingly, equilibria may also be found by seeking flows that are invariant under a single very large Backwards-Euler Forwards-Euler timestep. We show that this method, called Stokes preconditioning, is 10 to 50 times faster at computing steady states in plane Couette flow and traveling waves in pipe flow. Moreover, it can be carried out using Channelflow (by Gibson) and Openpipeflow (by Willis) without any changes to these popular spectral codes. We explain the convergence rate as a function of the integration period and Reynolds number by computing the full spectra of the operators corresponding to the Jacobians of both methods.

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Authors (presenter in bold)

Marta Net, UPC
Juan Sanchez Umbria, UPC

Abstract

Time periodic flows of low Prandtl number pure and binary fluids, filling slots heated by the side, are studied. To obtain the numerical solutions, the functions are approximated by a pseudo-spectral collocation method on a mesh of Gauss-Lobatto points. To discretize the momentum, energy and concentration equations, the spatial differential operators are transformed into matrices operating on the values of the functions at the collocation mesh points. The stiff system of ordinary differential equations, obtained after the spatial discretization, is integrated by means of fifth order semi-implicit backward-differentiation-extrapolation formulas. In the computation of the explicit terms, the differential operators at the mesh points are approximated by matrix-matrix products, using a high-performance implementation of the {DGEMM} subroutine of the {BLAS} library. The linear systems to be solved in the implicit part use fast Poisson solvers. Continuation techniques and stability analysis of the solutions are then applied. The bifurcation points on the branches of solutions are determined with precision by calculating their leading spectra, for a large range of Rayleigh and Prandtl numbers. The role played by the shear stresses of the steady field and the two buoyancies, at the onset of the oscillations, is analyzed by means of the energy equation for the perturbations, and compared with the contour plots of the eigenfunctions at the bifurcating points.

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Authors (presenter in bold)

Hugh M. Blackburn, Monash University
Philip Hall, Monash University

Abstract

We describe the methodology and application of a hybrid spectral element/asymptotic approach to computation of doubly periodic self-sustaining vortex-wave solutions to Navier-Stokes problems in an infinite shear flow.

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Authors (presenter in bold)

Andrew L. Hazel, The University of Manchester
Andres Franco-Gomez, The University of Manchester
Alice B. Thompson, The University of Manchester
Anne Juel, The University of Manchester

Abstract

We study the propagation of finite air bubbles in Hele-Shaw channels, rigid channels with rectangular cross sections of large aspect ratio, driven by constant volumetric flow rate of a surrounding viscous fluid. Introducing a centred, axially-uniform depth constriction adds considerable complexity to dynamics of the system. We restrict attention to flows in which the fluid inertia is negligible, but the presence of constant surface-tension at the air-fluid interface leads to nonlinear behaviour.

In the absence of the constriction, bubbles always migrate to the centre of the cross-section, the only stable steadily-propagating solution. If a constriction is present the stability of the centrally propagating mode depends on the flow rate; and alternative propagation modes where the bubble propagates within one of the deeper side-channels (away from the constriction) can arise through symmetry-breaking bifurcations from the centrally propagating solution.

We use a combination of high-precision experiments and adaptive finite element simulations of a depth-averaged model to examine the possible propagation modes and their stability. The simulations are conducted using oomph-lib (http://www.oomph-lib.org) which includes tools for parameter continuation and bifurcation detection.

The physical mechanisms that underly the observed complex behaviour result from a subtle interaction between viscous and surface tension forces, each of which can be stabilising or destabilising depending on the bubble morphology, location and flow rate. The depth-averaged model accurately predicts all the steady propagation modes observed experimentally, and provides a comprehensive picture of the underlying steady bifurcation structure. We find that the geometry and flow rates can be tuned such that whether the bubble is centred or off-centred depends on its size, which has potential applications in passive bubble sorting devices.

From a fundamental point of view, the constriction provokes interaction between the countably infinite solutions of the uniform Hele-Shaw system and (weakly) unstable solutions continue to play a role in the transient dynamics. For sufficiently large flow rates, initially centred bubbles do not converge onto a steady mode of propagation, but transiently explore the weakly unstable steady modes, an evolution which results in their break-up and eventual settling into a steady propagating state of changed topology.

Minisymposium Organisers

Daan Huybrechs, KU Leuven
Marcus Webb, KU Leuven, FWO Flanders

Abstract

Spectral methods have seen great success for solving differential equations on simple geometries such as intervals and tensor product domains. In this minisymposium, talks will demonstrate the new directions spectral methods have taken in recent years, to go beyond the simplest geometries and beyond the simplest choices of a spectral basis.

For example, efficiency and stability improves by utilising different bases at different steps in the discretisation, tailor-made so that the operators become highly structured. An expansion in Chebyshev polynomials may result in expansions in Gegenbauer or other orthogonal polynomials, in such a way that fast transforms can be devised, allowing spectral methods to scale to large numbers of unknowns.

Furthermore, a promising new direction is the introduction of redundancy into a discretisation. This leads to representations in a frame rather than a basis. The redundancy leads to great flexibility: frames enable discretisations on domains with complicated geometries, the inclusion of singular behaviour, and spectrally accurate representations of functions with distinct features not well captured by a single basis.

The minisymposium groups two talks in the former category of innovation, and two in the latter category.

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Authors (presenter in bold)

Sheehan Olver, Imperial College

Abstract

Multivariate orthogonal polynomials on the circle and sphere are classical: they are precisely Fourier series and spherical harmonics. The circle and sphere are special cases of algebraic curves/surfaces, that is, the zero set of a polynomial. This raises the following question: what are the properties of orthogonal polynomials on other algebraic curves? We will answer the question on two simple cases: a wedge and the boundary of the square, discuss general properties on other algebraic curves, and investigate some applications to PDEs and free boundary problems.

Joint work with Yuan Xu.

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Authors (presenter in bold)

Daan Huybrechs, KU Leuven
Ben Adcock, Simon Fraser University

Abstract

Spectral and high-order methods have always been difficult on domains with complicated shapes. The primary reason is the lack of a good basis, that offers numerical stability and is accompanied by a fast approximation algorithm. Bases with these properties in more than one dimension are known only for a very limited set of domains with special structure.

We show that this problem is entirely circumvented by considering more general function sets, and we focus on so-called frames. Unlike a basis, a frame can be redundant and this allows enormous flexibility. In particular, it is trivial to come up with frames for complicated domains. With such frames, smooth functions or even functions with known singular behaviour can be well approximated. The difficulty shifts to the approximation problem: how does one compute a frame approximation? Due to the redundancy, the linear systems involved have multiple solutions and, hence, are numerically ill-conditioned. We show that, among all possible redundant sets, frames are such that the approximation problem is, somewhat surprisingly, well-conditioned. Moreover, we indicate the possibility of fast algorithms for the spectral approximation of functions defined on domains with arbitrary shape.

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Authors (presenter in bold)

Geoff Vasil, University of Sydney
Keaton Burns, MIT
Daniel Lecoanet, Princeton University
Sheehan Olver, Imperial College London

Abstract

On a two-sphere (for example), not all smooth functions of latitude and longitude represent non-singular physical quantities; the concept of longitude breaks down at the north and south poles. The same kind of breakdown happens at the centres of the 2D disk or 3D ball. Truly well-behaved functions must contain special relationships between otherwise independent variables near coordinate singularities. Capturing these dependencies can become numerically very challenging. Fortunately, the famous Spherical Harmonic functions perfectly satisfy the special behaviour (for scalar functions) near the poles and can be used as a complete spectral basis. But what happens for vectors, higher-order tensors, and other geometries?

This talk will discuss how to represents functions, vectors, and arbitrary tensors on domains with spherical symmetry and coordinate singularities in an accurate and numerically efficient way. The solution is based on the remarkable properties of Jacobi polynomials; which include Legendre and Chebyshev polynomials as special cases, and Hermite and Laguerre as limiting cases. Using Jacobi polynomials, it is possible to construct basis sets that account for a wide range of singular behaviour. These bases inherit simple and sparse differentiation and multiplication formula from the algebra of Jacobi polynomials, which allows for the efficient solution of PDEs.

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Authors (presenter in bold)

Marcus Webb, KU Leuven, FWO Flanders

Abstract

This talk introduces new spectral methods for differential equations, which use Fourier extensions. A Fourier extension is a representation of a function on an arbitrarily shaped domain by a Fourier series which is periodic on a hypercuboid containing said domain. Such representations are extremely ill-conditioned, because the values of the approximant in the complement of the domain are disregarded for the approximation. Nonetheless, Fourier extensions provide spectrally convergent approximations of functions on a complicated shape, and furthermore, these approximations can be computed in a fast and stable manner in spite of the ill-conditioning (see the talk of Daan Huybrechs in this minisymposium). We show how these recent methods for function approximation can be adapted to build spectral methods which have the same flop-complexity.

Minisymposium Organisers

Ana Alonso Rodriguez, University of Trento
Francesca Rapetti, Université Cote Azur

Abstract

Electromagnetic phenomena is one of the foundations of modern technology and the development of highly efficient approximation methods for electromagnetic computations has became a strengthened research field on computational mathematics. Our aim is to discuss different high order and high performance methods that can be used for low frequency electromagnetic simulations. We would like to cover a wide range of approaches: from methods well established in computational electromagnetism, like high order Whitney forms, discontinuous Galerkin or domain decomposition methods, to more recent approaches like isogeometric analysis, reduced basis methods or virtual elements, that are a novelty (or even not yet applied) in this kind of problems.

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Authors (presenter in bold)

Marcella Bonazzoli, Inria, Sorbonne Université, Paris

Abstract

The finite element approximation of problems arising in computational electromagnetism requires spaces of vectors with polynomial components, chosen in such a way that the approximated field satisfies conformity conditions more complicated than for nodal finite elements. More precisely, finite element subspaces of H_curl, respectively H_div, are characterized by the continuity of the tangential, respectively normal, component of the approximated field. The practical implementation of these finite elements is particularly delicate in the high order case, especially in 3d. Thus we present a revisitation of classical degrees of freedom well-suited to implementation and a technique to restore duality between generators and degrees of freedom. Moreover, in applications this finite element discretization yields large scale linear systems, which can be severely ill conditioned, so that preconditioning becomes mandatory for iterative solvers.

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Authors (presenter in bold)

Lourenco Beirao da Veiga, Università Milano Bicocca
Franco Brezzi, IMATI
Franco Dassi, Università Milano Bicocca
Donatella L. Marini, Università di Pavia, IMATI del CNR
Alessandro Russo, Università Milano Bicocca

Abstract

We consider the variational formulation of magnetostatic problems proposed by Kikuchi, where the magnetic field, $\mathbf{H}$, is the unknown vectorial function.

We depart from the standard Finite Element Method (FEM) and we apply for the first time the Virtual Element Method (VEM) to solve such problems. The Virtual Element Method is a novel way to discretize a partial differential equation. It avoids the explicit integration of shape functions and introduces an innovative construction of the stiffness matrix so that it acquires very interesting properties and advantages. One among them is the possibility to apply the VEM to general polygonal/polyhedral domain decomposition, also characterized by non-conforming and non-convex elements.

In this talk we focus on the definition/construction of the lowest degree elements to better appreciate the novelty of VEM. Then, we give an hint of the general order case. Finally, we show some numerical examples to numerically validate the theoretical analysis of the method.

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Authors (presenter in bold)

Rafael Vazquez, Ecole Polytechnique Federale de Lausanne, IMATI-CNR
Annalisa Buffa, Ecole Polytechnique Federale de Lausanne

Abstract

Isogeometric methods consist in the discretization of partial differential equations using functions utilized in Computer Aided Design, such as NURBS or B-splines, for the approximation of the discrete solution. Structure preserving isogeometric methods were first introduced in [1], and have been successfully used in fluid mechanics and computational electromagnetism afterwards. The methods developed in [1] are based on a De Rham complex of B-spline spaces, that can be seen as a generalization of edge and face finite elements with higher continuity across elements.

In the context of finite elements, a dual finite element complex, defined in a new mesh constructed by barycentric refinement, was introduced by Buffa and Christiansen in [2]. In the present paper we develop a dual spline complex for isogeometric methods. Compared to the dual complex in [2], the dual spline complex we present is much easier to construct, thanks to the tensor-product structure of B-splines. We will also show that the construction can be generalized to domains formed by the union of several patches, generalizing the tensor-product construction to arbitrary topology.

[1] A. Buffa, G. Sangalli, R. Vazquez, Isogeometric analysis in electromagnetics: B-splines approximation, Comput. Methods Appl. Mech. Engrg. 199 (2010), 1143-1152.

[2] A. Buffa and S. Christiansen, A dual finite element complex on the barycentric refinement, Math. Comp. 76 (2007) 1743-1769.

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Authors (presenter in bold)

Blanca Ayuso de Dios, Universita di Bologna
Ralf Hiptmair, ETH Zurich
Cecilia Pagliantini, École Polytechnique Fédérale de Lausanne

Abstract

Discontinuous Galerkin (DG) discretizations of eddy current problems in the presence of moving fluids offer great flexibility in dealing with different flow regimes and incorporating discontinuities of the medium. The severe time-step restrictions associated with high-order polynomial discretizations motivate the use of semi-implicit timestepping for which efficient solvers become compulsory. We develop a family of preconditioners for the linear system of equations ensuing from piecewise polynomial symmetric Interior Penalty DG discretizations of H(curl)-elliptic boundary value problems on conforming meshes. The design and analysis of the proposed preconditioners relies on an auxiliary space of H(curl)-conforming finite element functions together with a relaxation technique (local smoothing). We discuss the asymptotic convergence of the resulting preconditioners and the robustness of the solvers with respect to jumps in the coefficients of the second and zero-th order parts of the operator.

Minisymposium Organisers

Vianey Villamizar, Brigham Young University, Department of Mathematcis
Otilio Rojas, Barcelona Supercomputing Center, Department of Computer Applications in Science and Engineering

Abstract

This minisymposium is concerned with new developments of high order methods for acoustic, electromagnetic, and elastodynamic problems in wave propagation. The participants will address challenging scientific and mathematical problems in the propagation and scattering of waves in increasingly complex settings. The numerical techniques considered includes: Isogeometric Analysis, Finite Elements, and Finite Differences. There is a special emphasis on coupling high order methods with new local and high order absorbing boundary conditions for unbounded problems. The applications range from geophysical simulations to multiple acoustic scattering problems.

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Authors (presenter in bold)

Michael Medvinsky, NC State University
Semyon Tsynkov, NC State University
Eli Turkel, Tel Aviv University

Abstract

We consider exterior problems for the three-dimensional Helmholtz equation. The computational domain is terminated by a spherical artificial boundary where we set local high order artificial boundary conditions (ABCs) of Bayliss-Gunzburger-Turkel type (BGT). The key new result that we present is that the BGT conditions are imposed directly and do not require any auxiliary variables to be introduced at the artificial outer boundary.

The Helmholtz equation is discretized with sixth order accuracy on a Cartesian grid using a compact finite difference scheme. The non-conforming spherical outer boundary is handled by the method of difference potentials (MDP), which guarantees no deterioration of accuracy.

The BGT boundary conditions are built as a sequence of successively more accurate approximations, each subsequent boundary condition canceling the next term in the far field expansion of the solution. The BGT condition of a given order involves a differential operator that contains the first normal (i.e., radial) derivative, as well as higher order tangential derivatives. Previously, the BGT conditions could be implemented directly only up to second order. Higher order BGT conditions relied on auxiliary variables, as first proposed by Hagstrom and Hariharan. This allowed one to avoid the discretization of higher order tangential derivatives (polar and azimuthal).

However, the tangential derivatives in BGT are not arbitrary; they come as powers of the Beltrami operator on the sphere plus additional first order derivatives with respect to the polar and azimuthal angle. This suggests using the eigenfunctions of the Beltrami operator as the continuous basis functions required by the MDP. These eigenfunctions are the well-known spherical harmonics. On one hand, they provide a convenient orthogonal basis for the MDP. On the other hand, they eliminate the tangential derivatives beyond the first order, since the powers of the Beltrami operator are replaced with the powers of the corresponding eigenvalues that are available analytically.

The proposed methodology is tested numerically. The simulations corroborate its design level of performance.

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Authors (presenter in bold)

Fouche Smith, NC State University
Semyon Tsynkov, NC State University
Eli Turkel, Tel Aviv University

Abstract

We build a compact fourth order accurate finite difference scheme, in both space and time, for the three-dimensional wave (d'Alembert) equation with variable propagation speed. The resulting scheme is implicit in time; it uses a 3x3x3 stencil in space on three consecutive time levels. A compact stencil means that the scheme requires no auxiliary boundary conditions beyond those needed by the underlying differential equation itself. The scheme is implemented by forming a right-hand side composed of the solution on the previous two time levels and then solving a modified Helmholtz equation (negative definite) to obtain the solution on the upper time level.

In the simplified (yet important) case of a constant propagation speed, the inversion at the upper time level is done by FFT. Then, the complexity of the resulting time-marching procedure is essentially the same as that of an explicit scheme. In the more general case of a variable propagation speed, the inversion is done by multigrid. The required number of iterations can be estimated using local Fourier/eigenvalues analysis. In practice, if one takes the solution from the previous time level as the initial guess, only a small number of multigrid cycles proves sufficient for driving the error of the iterative solution down to the level of the discretization error on the grid. The overall complexity remains comparable to that of an explicit method.

Numerical simulations corroborate the design fourth order accuracy of the scheme for both the constant coefficient wave equation and variable coefficient wave equation in 3D. Altogether, even though the proposed scheme is implicit and only conditionally stable, it is still considerably more efficient than its lower order counterparts.

The current work extends our previous development done for the case of two space dimensions.

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Authors (presenter in bold)

Sebastian Acosta, Texas Children's Hospital, Baylor College of Medicine

Abstract

We present a new family of high order on-surface radiation conditions to approximate the solution to the Helmholtz equation in exterior domains. Motivated by the pseudo-differential expansion of the Dirichlet-to-Neumann map developed by Antoine et al. (1999), we design a systematic procedure to apply pseudo-differential symbols of arbitrarily high order. Numerical results are presented for solving both the Dirichlet and the Neumann boundary value problems. Possible improvements, extensions and use as preconditioner will be discussed.

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Authors (presenter in bold)

Tahsin Khajah, University of Texas at Tyler
Vianey Villamizar, Brigham Young University

Abstract

This work is concerned with a unique combination of high order local absorbing boundary conditions and a highly accurate IsoGeometric finite element Analysis (IGA) used in the interior of the computational domain for time-harmonic acoustic waves. We employ absorbing boundary conditions derived from exact Farfield Expansions representations of the outgoing waves in the exterior of the regions enclosed by the artificial boundaries. It is shown that IGA exhibits a superior spectral approximation, higher convergence rate per degree of freedom, and lower pollution error when compared with conventional finite element method. Furthermore, IGA accurately models complex geometries using relatively coarse discretizations, which lowers the computational cost when combined with high order basis functions.

In addition, we demonstrate that the proposed method can reduce both the pollution and artificial boundary errors to negligible levels even in mid- to high- frequency analysis with rather course discretization densities. As a result, we have developed a highly accurate computational platform to numerically solve acoustic wave scattering in 2D and 3D. The results of our numerical experiments reveal that the numerical error is no longer limited by the error incurred in the approximation at the artificial boundary. In fact, the error on the artificial boundary can be reduced conveniently such that the dominant error comes from the volume discretization method used in the interior of the computational domain.

Minisymposium Organisers

Misun Min, Argonne National Laboratory
Paul Fischer, University of Illinois, Argonne National Laboratory
Tzanio Kolev, Lawrence Livermore National Laboratory

Abstract

High-order discretizations have the potential to provide an optimal strategy for achieving high performance and delivering fast, efficient, and accurate simulations on next-generation architectures. Traditional low-order schemes, developed in an era when FLOPs were the performance-limiting factor, will no longer be an attractive option, since the cost of FLOPs will be considerably smaller than the cost of data motion. While many high-order methods were intractable on older architectures, their importance and usefulness has been dramatically increasing and has demonstrated significant improvements in both simulation quality and parallel strong scalability.

This session will discuss the next-generation high-order discretization algorithm and software, based on finite/spectral element approaches that will enable a wide range of important scientific applications to run efficiently on future architecture.

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Authors (presenter in bold)

Peter Bastian, Heidelberg University
Dominic Kempf, Heidelberg University
Eike Mueller, University of Bath
Steffen Müthing, Heidelberg University
Marian Piatkowski, Heidelberg University

Abstract

Matrix-free implementation of high-order finite element methods using sum factorization offers the possibility bypassing the memory bottleneck while at the same time reducing the number of operations substantially. In this talk we particularly address three issues arising with this approach. 1) How to achieve a substantial fraction of peak-performance of modern multi-core processors with wide SIMD instructions through a combination loop fusion and splitting techniques. 2) How to achieve performance portability across different PDEs and hardware platforms using code generation techniques. 3) How to effectively precondition elliptic problems with highly varying coefficients using a combination of matrix-free block smoother and low-order subspace correction. We will present numerical results of our implementation within the DUNE-PDELAB software framework on scalar problems and systems of PDEs as well as on various hardware platforms including AVX512.

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Authors (presenter in bold)

Martin Kronbichler, Technical University of Munich

Abstract

My talk will present the framework for matrix-free evaluation of finite element operators contained in the deal.II finite element library, relying on sum factorization specialized for quadrilateral and hexahedral meshes. The kernels target the case of fast quadrature for both cell as well as face integrals and are applicable to a wide class of weak forms originating from linear and nonlinear partial differential equations. The algorithms have been selected with the goal of achieving high application performance on modern cache-based architectures, guided by performance analysis. The central ingredients are vectorization over several cells or faces based on C++ wrapper classes around instrinsics and an even-odd decomposition of the one-dimensional compute kernels that further cuts operations into half for moderate and high polynomial degrees. In isolation, the compute kernels reach up to 60% of arithmetic peak on Intel Haswell and Broadwell processors and up to 50% of arithmetic peak on Intel Knights Landing. The high degree of compute optimization make the memory transfer in loading the input and output vectors, MPI ghost exchange, as well as handling variable coefficients and the geometry the most critical factor, in particular on CPUs with moderate memory bandwidth. Furthermore, the throughput is affected by indirect addressing in gather and scatter instructions that appear in the cell integrals of globally continuous function spaces or the face integrals of fully discontinuous function spaces.

These code components have shown outstanding performance in the simulation of incompressible fluid flow and wave propagation. My talk will highlight some of the downstream solver components based on the matrix-free operator evaluation, including fast multigrid solvers for high-order methods that have been run on up to 150,000 cores. In particular, I will present our group's work on identifying suitable polynomial bases that maximize the efficiency of operator evaluation depending on the application setting.

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Authors (presenter in bold)

Noel Chalmers, Virginia Tech
Tim Warburton, Virginia Tech

Abstract

In this talk we detail a new preconditioning technique for high-order finite element methods on triangular elements. The formulation of the preconditioner is based on replacing the high-order approximation with a low-order version which captures the spectral properties of the high-order operators. Classical low-order preconditioning approaches [Orszag 1980] have been demonstrated to work well in one-dimensional and tensor-product bases, but not for triangular elements. We propose a new formulation, where the nodes of the low-order finite element problem do not necessarily coincide with the high-order nodes. Instead, the two spaces are connected using least squares projection operators. Our computational results show that the condition number of the preconditioned high-order stiffness matrix on the reference element can be well-controlled and the best performing preconditioners are formed with low-order finite element meshes which have more degrees of freedom than the high-order approximation with nodes grouped close to the element edges. We also present computational results which show that the performance of algebraic multigrid (AMG) preconditioners is improved at high-order by preconditioning the low-order rather than the high-order approximation.

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Authors (presenter in bold)

Jed Brown, CU Boulder
Jeremy Thompson, CU Boulder
Jean-Sylvain Camier, Lawrence Livermore National Laboratory
Tzanio Kolev, Lawrence Livermore National Laboratory
Veselin Dobrev, Lawrence Livermore National Laboratory
Thilina Rathnayake, University of Illinois, Urbana-Champaign

Abstract

One of the challenges with high-order finite element and spectral element methods is that a global sparse matrix is no longer a good representation of a high-order linear operator, both with respect to the FLOPs needed for evaluation and the memory transfer needed for a matrix-vector multiply. Thus, high-order methods require a new operator description that still represents a linear (or more generally non-linear) operator. libCEED is an extensible library that provides a portable algebraic interface and optimized implementations suitable for high-order operators. libCEED's operator description is easy to incorporate in a wide variety of applications, without significant refactoring of the discretization infrastructure. We introduce the libCEED API and compare performance to native implementations in production software, such as Nek5000 and MFEM.

Minisymposium Organisers

Philipp Schlatter, KTH Royal Institute of Technology

Abstract

The complex nature of turbulent fluid flows implies that the computational resources needed to accurately model problems of industrial and academic relevance is virtually unbounded. Computational Fluid Dynamics (CFD) is therefore a natural driver for exascale computing both for academic and industrial cases, and has the potential for substantial societal impact, like reduced energy consumption, alternative sources of energy, improved health care, and improved climate models. This proposed minisymposium will be organized by the EU funded Horizon 2020 project ExaFLOW and will feature presentations showcasing their work on addressing key algorithmic challenges in CFD in order to facilitate simulations at exascale, e.g. accurate and scalable solvers, data reduction methods and strategies to ensure fault tolerance and resilience.

In particular, the talks in this minisymposium will highlight the following topics:

Adaptive mesh refinement: The generation of a high-quality mesh is a complex task and is usually considered as a bottleneck for any user of a CFD code. Ideally, the mesh should avoid unnecessary over-resolution in unimportant regions, resolve the critical parts and still require as few computational resources as possible. Adaptive mesh refinement (AMR) techniques have been developed to tackle this issue by, first, identifying poorly resolved regions with effective error estimators based on adjoints, and then, applying a mesh refinement strategy. Both techniques are implemented in Nek5000, an open-source, highly scalable and portable code based on the spectral element method (SEM). Issues of efficient implementation of non-conformal elements, preconditioners and potential stability issues will be discussed, together with an analysis of adjoint-based error estimators in viscous flow. Results for 2D and 3D laminar and turbulent flows are presented.

Mixed CG-HDG schemes: The number of computing cores expected to become available in an exascale environment means that there is huge potential for significantly decreasing the execution times of large-scale CFD simulations. However to capitalise on this, we require methods that have good strong-scaling potential. The strong scaling of high-order methods, whilst typically better than lower-order schemes, is still expected to be an issue for even extremely large simulations. We will show progress in deriving a mixed scheme of continuous Galerkin (CG) and hybridizable discontinuous Galerkin (HDG) methods in order to reduce communication between nodes whilst retaining favorable on-node performance.

Leveraging high-order methods for industrial flows: Another key impact area for high-order methods is their potential to give new insights into complex industrial flows. This is particularly true as we look towards leveraging exascale platforms to perform high resolution LES/DNS experiments, which are yet to be used as part of the industrial design process over existing RANS/DES methods due to the computational requirements. This presentation will highlight the progress being made in applying developments under the ExaFLOW project to investigate key aerodynamic challenges in the Formula 1 motorsport industry.

This session aims at bringing together the CFD community as a whole, from HPC experts to domain scientists, discuss current and future challenges towards exascale fluid dynamics simulations and facilitating international collaboration.

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Authors (presenter in bold)

Adam Peplinski, KTH Royal Institute of Technology
Nicolas Offermans, KTH Royal Institute of Technology
Paul Fischer, University of Illinois
Philipp Schlatter, KTH Royal Institute of Technology

Abstract

Nek5000 is based on the spectral-element method (SEM), where the computational domain is decomposed into a set of disjoint sub-domains (elements). This domain decomposition is a main source of parallelism, but also provides flexibility in grid generation that can be used in Adaptive Mesh Refinement (AMR), which controls computational error during the simulation by placing higher resolution grid where it is needed. AMR gives possibility to increase the accuracy of numerical simulations at minimal computational cost, but at the same time it increases solver complexity, that can have negative effect on the code performance. There are three possible AMR approaches for SEM code. Two of them modify number of degrees of freedom by splitting/merging elements (h-refinement) or by local variation of the polynomial order inside particular element (p-refinement). On the contrary the last approach keeps constant number of degrees of freedom and modifies resolution by redistribution of grid points. Within ExaFLOW project we work on h-type AMR framework for Nek5000.

This strategy dynamically changes element number and global mesh connectivity, and requires dynamical mesh manager and mesh partitioner for proper load balancing. The grid refinement and de-refinement is managed by p4est library enabling the dynamic management of a collection of adaptive octrees. This library provides element connectivity information for the dual graph, which is later manipulated by ParMETIS giving new element to processor mapping. Developing nonconforming version of Nek5000 we used geometrically nonconforming and functionally conforming approach based on the previous work of Fischer, Kruse and Loth. As the solver complexity grows special care has been taken to develop efficient tools that can be used within AMR framework, in particular to proper adaptation of pressure preconditioners for nonconforming SEM. In this presentation we discuss implementation of h-type AMR and related stability issues for SEM solvers. We present as well a few test cases.

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Authors (presenter in bold)

Nicolas Offermans, KTH Royal Institute of Technology
Adam Peplinski, KTH Royal Institute of Technology
Oana Marin, Argonne National Laboratory
Philipp Schlatter, KTH Royal Institute of Technology

Abstract

Simulations using adaptive mesh refinement (AMR) are performed in Nek5000, a code aimed at direct numerical simulations, based on the spectral element method (SEM). The two main ingredients for AMR are tools for automatic mesh refinement and error estimators which allow for optimal error control. New capabilities in Nek5000 enable the use of the h-refinement method for mesh adaptation, where selected elements are split via quadtree (2D) or octree (3D) structures. Two methods are considered for estimating the error. The first method is local and based on the spectral properties of the solution on each element. These so-called spectral error indicators are based on a method developed by Mavriplis, they come with a low overhead and are easily implemented but they provide only a local measure of the error. The second method, on the other hand, is goal-oriented and takes into account both the local properties of the solution and the global dependence of the error in a functional of interest (such as drag or lift) via the resolution of an adjoint problem. These so-called adjoint error estimators are based on a similar work done within the framework of the finite element method by Johnson and Hansbo or Bangerth and Rannacher.

AMR capabilities are demonstrated in Nek5000 and applied to simple steady and unsteady test cases, such as the flow past a cylinder and the lid-driven cavity, in two and three dimensions. The definition and implementation of both error estimation methods are presented and discussed. The results obtained with both methods are compared and are shown to efficiently reduce the number of degrees of freedom required to reach a given tolerance on the solution compared to conforming refinement. Moreover, the gains in terms of mesh generation, accuracy and computational cost are discussed by analyzing the convergence of some functional of interest and the evolution of the mesh as refinement proceeds.

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Authors (presenter in bold)

Martin Vymazal, Imperial College London
David Moxey, University of Exeter
Chris Cantwell, Imperial College London
Robert M. Kirby, University of Utah
Spencer Sherwin, Imperial College London

Abstract

We combine continuous and discontinuous Galerkin methods in the setting of a model diffusion problem. Starting from a hybrid discontinuous formulation, we replace element interiors by more general subsets of the computational domain - groups of elements that support a piecewise-polynomial continuous expansion. This step allows us to identify a new weak formulation of Dirichlet boundary condition in the continuous framework. We show that the boundary condition leads to a stable discretization with a single parameter insensitive to mesh size and polynomial order of the expansion. The formal advantage of the new approach is that the weak formulation can be written as a sum of a discrete interior operator and one or more additional terms that enforce boundary constraints as is typical for Neumann boundary conditions. When tested on a scalar elliptic problem, the weak boundary condition is slightly less efficient than Nitsche's method on isotropic meshes, but is significantly more accurate and robust in the anisotropic case. We also demonstrate that the boundary conditions preserve the accuracy of derived quantities evaluated on wall boundaries such as lift and drag. Since our framework allows to combine continuous and hybrid discontinuous Galerkin methods, we discuss the performance implications of such setting.

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Authors (presenter in bold)

Evelyn Otero, KTH Royal Institute of Technology
Ricardo Vinuesa, KTH Royal Institute of Technology
Oana Marin, Argonne National Laboratory
Erwin Laure, KTH Royal Institute of Technology
Philipp Schlatter, KTH Royal Institute of Technology

Abstract

Postprocessing and storage of large data sets represent one of the main computational bottlenecks in computational fluid dynamics. We assume that the accuracy necessary for computation is higher than needed for postprocessing. Therefore, in the current work we assess thresholds for data reduction as required by the most common data analysis tools used in the study of fluid flow phenomena, specifically wall-bounded turbulence. These thresholds are imposed a priori by the user, and we assess a set of parameters to identify the minimum accuracy requirements. The method considered in the present work is the discrete Legendre transform (DLT), suitable for the spectral element method. We evaluate it in the computation of turbulence statistics, spectral analysis and resilience for cases highly-sensitive to the initial conditions. Maximum acceptable compression ratios of the original data have been found to be around 97%, depending on the application purpose. The new method outperforms straightforward downsampling, as well as the previously explored data truncation method based on discrete Chebyshev transform (DCT). For spectral elements, all the computational work is completely local, and no communication is needed during compression or decompression.

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Authors (presenter in bold)

Chris Cantwell, Imperial College London
Allan Nielsen, École polytechnique fédérale de Lausanne

Abstract

We describe a novel, minimally intrusive approach to adding fault tolerance to existing complex scientific simulation codes, used for addressing a broad range of time-dependent problems on the next generation of supercomputers. Exascale systems have the potential to allow much larger, more accurate and scale-resolving simulations of transient processes than can be performed at on current petascale systems. However, with a much larger number of components, exascale computers are expected to suffer a node failure every few minutes. Many existing parallel simulation codes are not tolerant of these failures and existing resilience methodologies would necessitate major modifications or redesign of the application. Our approach combines the proposed user-level failure mitigation extensions to the Message-Passing Interface (MPI), with the concepts of message-logging and remote in-memory checkpointing, to demonstrate how to add scalable resilience to transient solvers. Logging MPI communication reduces the storage requirement of static data and allows a spare MPI process to rebuild these data structures independently. Remote in-memory checkpointing avoids disk I/O contention on large parallel filesystems. A prototype implementation is applied to Nektar++, a scalable, production-ready transient simulation framework. Forward-path and recovery-path performance of the resilience algorithm is analysed through experiments using the solver for the incompressible Navier-Stokes equations, and strong scaling of the approach is observed.

Minisymposium Organisers

Yulong Xing, Ohio State University
Xiangxiong Zhang, Purdue University

Abstract

The focus of this special session will be on recent developments in the design, analysis, implementation and application of structure preserving numerical methods, such as positivity/bound preserving, energy preserving, vorticity preserving, entropy satisfying, well-balanced and asymptotic preserving methods. The special session aims to bring together researchers from various fields to report the recent developments in structure preserving methods and to discuss the relevant applications. The topics will span a range from theoretical results to novel algorithms, and will include talks focused on a variety of interesting application areas.

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Authors (presenter in bold)

Huazhong Tang, Peking University, Xiangtan University

Abstract

We first study the admissible state set G of special relativistic magnetohydrodynamics (RMHD). It paves a way for developing physical-constraints-preserving (PCP) schemes for the special RMHD equations with the solutions in G. To overcome the difficulties arising from the extremely strong nonlinearities and no explicit formulas of the primitive variables and the flux vectors with respect to the conservative vector, two equivalent forms of G with explicit constraints on the conservative vector are skillfully discovered. The first is derived by analyzing roots of several polynomials and transferring successively them, and further used to prove the convexity of G with the aid of semi-positive definiteness of the second fundamental form of a hypersurface. While the second is derived based on the convexity, and then used to show the orthogonal invariance of G. The LxF splitting property does not hold generally for the nonzero magnetic field, but by a constructive inequality and pivotal techniques, we discover the generalized LxF splitting properties, combining the convex combination of some LxF splitting terms with a discrete divergence-free condition of the magnetic field. Based on the above analyses, several 1D and 2D PCP schemes are then studied. In the 1D case, a first-order accurate LxF-type scheme is first proved to be PCP under the CFL condition, and then the high-order accurate PCP schemes are proposed via a PCP limiter. In the 2D case, the discrete divergence-free condition and PCP property are analyzed for a first-order accurate LxF-type scheme, and two sufficient conditions are derived for high-order accurate PCP schemes. Our analysis reveals in theory for the first time that the discrete divergence-free condition is closely connected with the PCP property. Several numerical examples demonstrate the theoretical findings and the performance of numerical schemes.

References: [1] K.L. Wu and H.Z. Tang, Mathematical Models and Methods in Applied Science, 27 (2017), 1871-1928. [2] K.L. Wu and H.Z.Tang, Astrophysical Journal Supplement series, 228(1), 2017, 3. [3] K.L. Wu and H.Z. Tang, J. Comput. Phys., 298(2015), 539-564. [4] K.L. Wu and H.Z. Tang, arXiv: 1709.05838, 2017. [5] K. Wu, Phys. Rev. D, 95(2017), 103001.

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Authors (presenter in bold)

Hao Li, Purdue University
Shusen Xie, Ocean University of China
Xiangxiong Zhang, Purdue University

Abstract

We demonstrate that the classical fourth order accurate compact finite difference scheme satisfies a natural weak monotonicity property for solving scalar convection-diffusion equations. Based on such a property, we design a simple limiter to enforce the bound-preserving or positivity-preserving property of numerical solutions without losing conservation or high order accuracy. Higher order schemes including 6th and 8th order schemes satisfying the weak monotonicity can also be constructed. We show that the bound-preserving property is still valid when a total-variation-bounded (TVB) limiter is used to reduce oscillations. All these results can be easily extended to higher dimensions and passive convection such as incompressible flows. More general boundary conditions such as the inflow-outflow boundary condition will also be discussed. This is a joint work with Prof. Shusen Xie at Ocean University of China and Prof. Xiangxiong Zhang at Purdue University.

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Authors (presenter in bold)

Luc Grosheintz-Laval, ETH Zürich

Abstract

Hydrostatic equilibrium is a common, stationary, non-constant (in space) solution of the Euler equations with gravity. Since the solution is non-constant, classical numerical schemes will not preserve the equilibrium numerically. We will present a high-order FVM-based numerical scheme which preserves hydrostatic equilibrium to machine precision. Our scheme is not restricted to any specific equation of state, incl. tabulated, or to any specific gravitational potential.

Furthermore, we will present our advances on well-balanced schemes for moving equilibria, i.e. non-stationary equilibria, such as fluids rotating around an rotation axis. Here the force balance is between the centrifugal force, the pressure gradient and gravity.

We've implemented the well-balanced schemes in a parallel CPU-code on Cartesian grid and in a three-dimensional, multi-GPU code, called Tyr, on an unstructured grid of a spherical shell. Using these two codes, we will demonstrate the effectiveness of the algorithms on astrophysical examples from stellar evolution and core collapse super-nova.

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Authors (presenter in bold)

Sebastian Noelle, RWTH Aachen University
Yogiraj Mantri, RWTH Aachen University
Leonie Riebesam, RWTH Aachen University
Guoxian Chen, Wuhan University
Yulong Xing, Ohio State University

Abstract

The occurrence of vacuum is a notorious difficulty in continuum mechanics, which affects the modeling, the mathematical analysis, and the design of stable numerical algorithms. When computing shallow water flows with conservative finite volume or discontinuous Galerkin schemes, infinitesimal quantities may occur in momentum and mass. The velocity is obtained by dividing the two, which is a De L'Hospital type limit. This often leads to erratic front velocities, which subsequently affect the CFL condition and hence the time step. We are currently developing a new space-time quadrature near the wet-dry front. For some first cases, we can guarantee accurate front-velocities within the physically reasonable bounds.

Minisymposium Organisers

Gustaaf Jacobs, San Diego State University
Wai-Sun Don, School of Mathematical Sciences, Ocean University of China
David Kopriva, San Diego State University

Abstract

High order linear methods — such as spectral methods, optimized compact schemes, optimized central finite difference schemes, and discontinuous Galerkin methods — are widely used for solving PDEs with smooth solutions because of their efficient resolution and computation of both large and fine scale structures, including those existing in turbulence. However, the methods are usually applied, in practice, when the flow is marginally resolved or in the presence of shocks or high gradients that arise from the well-known Gibbs phenomenon. In contrast, schemes that regularize and stabilize — such as the weighted essentially non-oscillatory scheme (WENO), Discontinuous Galerkin with limiters and Spectral Domain Decomposition with physical based artificial viscosity— resolve discontinuities and high gradients in an essentially non-oscillatory manner, while also computing smooth structures with high resolution. In this minisymposium, we gather experts from the community to share their recent research and views with the goals of advancing high order methods to solve PDEs in marginal regimes and to discuss applications these methods can reliably compute.

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Authors (presenter in bold)

Philip Zwanenburg, McGill University
Siva Nadarajah, McGill University

Abstract

The Discontinuous Petrov-Galerkin (DPG) methodology was conceived with the goal of improving upon existing finite element methods through the use of partial differential equation (PDE)-dependent test spaces. The seminal work of Demkowicz et al. [1] provides the guiding principle that: while the \textit{trial} space must be chosen with good approximation properties, the \textit{test} space may be chosen solely to obtain good stability properties. While Discontinuous Galerkin (DG) methods add stabilization from the element boundaries through the prescription of numerical fluxes and traces, DPG methods leave both the numerical fluxes and traces as unknowns, seeking to find the ``correct'' values for these quantities automatically, using residual-based stabilization determined according to the minimization of the residual in a specified test norm.

Perhaps the most significant challenge with DPG methods is the choice of test norms for nonlinear PDEs. As a means of investigating suitable choices, we propose establishing a relation between physically motivated numerical fluxes/traces and the more abstract residual-based stabilization for various test norms, through the interpretation of the test space-induced stabilization as a corresponding upwinded differencing, in the same spirit as that performed for the SUPG formulation [2]. We will assess the effectiveness of the results based on the ability of the DPG formulation to outperform the standard DG formulation in terms of exhibiting higher-order convergence, such as for the Peterson test case [3], or improved stability, such as allowing for convergence on highly under-resolved meshes without requiring artificial diffusion or shock-capturing mechanisms.

We have begun our investigation with the implementation of the DPG method for the linear advection and Euler equations, but have the ultimate goal of establishing results for the Navier-Stokes equations.

References

[1]: L. Demkowicz, J. Gopalakrishnan, A class of discontinuous petrov-Galerkin methods. part i: The transport equation.

[2]: A. N. Brooks, T. J. Hughes, Streamline upwind/petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier-stokes equations.

[3]: T. E. Peterson, A note on the convergence of the discontinuous galerkin method for a scalar hyperbolic equation.

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Authors (presenter in bold)

David Kopriva, Florida State University, San Diego State University

Abstract

Practical implementation of a discontinuous Galerkin spectral element method (DGSEM) has the integrals in the formulation approximated with quadrature. For nonlinear, variable coefficient linear, or curved elements, the use of collocated quadratures leads to an approximation that is usually, but not always stable. To enhance the robustness of the schemes, the order of the quadratures can be increased leading to a procedure typically called ``overintegration''. Empirical evidence has suggested that overintegrating the volume terms of the approximation enhances robustness. Mengaldo et al. [G. Mengaldo, D. De Grazia, D. Moxey, P.E. Vincent, and S.J. Sherwin. Dealiasing techniques for high-order spectral element methods on regular and irregular grids. Journal of Computational Physics, 299:56 – 81, 2015.] showed that it is even better to also overintegrate the surface terms. Although it has been used in large codes such as Nektar++ and FLEXI, it is surprising that the stability of overintegration methods was not established theoretically.

In this talk we perform a linear analysis to show that aliasing errors introduced by the quadrature and the failure of the product rule to hold for interpolation approximations destabilizes the DGSEM. Over integration of the volume terms at a sufficiently high order can eliminate that destabilizing term, but introduces a new destabilizing term at the element boundaries that was not present in the under-integrated approximation. Overintegration of both the volume and surface integrals leads to a linearly stable approximation.

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Authors (presenter in bold)

Marvin Bohm, Universität zu Köln
Andrew R. Winters, Universität zu Köln
Gregor J. Gassner, Universität zu Köln
Dominik Derigs, Universität zu Köln
Florian Hindenlang, Max-Planck Institut

Abstract

In this talk an extension of discretely entropy stable discontinuous Galerkin (DG) methods to the resistive magnetohydrodynamics (MHD) equations is presented. In the continuous entropy analysis of the ideal MHD equations, which are the advective parts of the resistive MHD equations, the divergence-free constraint on the magnetic field components must be incorporated as a non-conservative term in a form either proposed by Powell or Janhunen. Consequently, this non-conservative term needs to be discretized, such that the approximation is consistent with the entropy.

Furthermore, as an extension of the ideal MHD system, the resistive parts are shown to satisfy the entropy inequality, continuously as well as discretely. This enables the construction of an entropy stable DG discretization for the resistive MHD equations.

However, the resulting method suffers from large errors in the divergence-free constraint, since no particular treatment of divergence errors is included in the standard resistive MHD model. Hence, we construct and analyze a DG method that is entropy stable for the resistive MHD equations and has a built-in generalized Lagrange multiplier (GLM) divergence cleaning mechanism. Moreover, the considered scheme is derived for unstructured three-dimensional curvilinear hexahedral meshes.

Finally, we provide numerical verification of the theoretical properties as well as the increased robustness of the derived method compared to standard high-order DG methods.

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Authors (presenter in bold)

Thomas Bolemann, University of Stuttgart, University of Cologne
David Flad, University of Stuttgart
Claus-Dieter Munz, University of Stuttgart

Abstract

Large Eddy Simulation has been a challenging field of research ever since, especially for high Reynolds numbers occurring in industrial applications. Conducting LES in these highly under-resolved settings necessitates the use of sub-grid scale turbulence models to make up for the lack of dissipation by the missing scales. There is consensus that a key requirement to build effective LES models is providing accurate and robust numerical schemes as a basis. Recent studies have shown that high-order numerical methods such as the discontinuous Galerkin or flux reconstruction method perform well for LES, due to their low numerical dissipation and dispersion errors, which help to avoid interaction between the models and the numerical approximation. However, in under-resolved settings the high-order schemes are prone to aliasing effects, which introduce numerical instabilities and affects the subgrid scale modelling.

A frequently used countermeasure for reducing aliasing errors is so-called polynomial de-aliasing, where a higher order quadrature rule is employed for evaluating the flux non-linearities. This strategy however, has two major shortcomings: First, computational cost increases due to the higher amount of quadrature points and, in case of discontinuous Galerkin spectral elements (DGSEM), the loss of collocation. Second, in severely under-resolved settings polynomial de-aliasing does not provide sufficient stabilization for compressible flows. An alternative can be found in the recently emerged split formulation DG schemes, which can be formulated to be kinetic energy preserving (KEP) and are thus extremely robust. This permits models to freely adjust the dispersion and dissipation characteristics, making the scheme ideal for LES.

In the talk we will briefly outline the basics of a split form for DGSEM, using a kinetic energy preserving flux formulation. We will show numerical results of established test cases for implicit and explicit LES, comparing KEP to conventional polynomial de-aliasing. We will furthermore present some recent work on LES modelling based on KEP schemes using explicit filters, where we investigate filter kernels optimized towards their dissipation properties.

Minisymposium Organisers

Henar Herrero, Universidad de Castilla-La Mancha
Juan Sánchez-Umbría, Universitat Politècnica de Catalunya

Abstract

The main objective of this minisymposium is to present some recent advances in high-order and spectral methods applied to hydrodynamic stability problems, where these methods play a relevant role due to their capability to produce very accurate numerical results. These problems include solving directly the Navier-Stokes equations, stability and eigenvalue problems, continuation of bifurcations techniques, etc.

Different numerical tecniques, computing implementations and applicationes will be presented, such as reduced basis, preconditioning, continuation, Rayleigh-Bénard, Taylor-Couette, bubble instabilities, ...

This minisymposium will bring together scientists to present their recent advances on algorithm development and analysis of high-order and spectral methods applied to hydrodynamic instability problems, provide a forum for discussion and interaction in these methods.

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Authors (presenter in bold)

Francisco Pla, Universidad de Castilla-La Mancha
Henar Herrero, Universidad de Castilla-La Mancha

Abstract

The reduced basis approximation is a discretization method that can be implemented for solving of parameter-dependent problems. This method consists of approximating the solution by a linear combination of appropriate preliminary computed solutions by an iterative procedure using the kolmogorov n-width measures [2, 4].

Rayleigh-Bénard convection problem displays multiple steady solutions and bifurcations by varying Rayleigh number R, therefore the eigenvalue problem of the corresponding linear stability analysis has to be implemented. A linear stability analysis of these solutions is performed in [3] by a spectral collocation method.

In this work the eigenvalue problem of the corresponding linear stability analysis is solved with the reduced basis method. It is considered the aspect ratio 3,495 and parameter R varies in [1000; 2000] where different stable and unstable bifurcartion branches appear [1, 3]. The reduced basis method is suitable to catch bifurcations and instabilities of stationary solutions for many values of R. The reduced basis considered belong to the eigenfunction spaces coming from the eigenvalue problems for different types of solutions in the bifurcation diagram. The eigenvalues and eigenfunctions are easily calculated and the bifurcation points are exactly captured. The resulting matrices are small and this allows a drastic reduction of the computational cost on the eigenvalue problems.

The problem is numerically solved by the Galerkin variational formulation using the Legendre Gauss-Lobatto quadrature formulas together with the reduced basis.

REFERENCES [1] H. Herrero, Y. Maday and F. Pla, RB (Reduced basis) for RB (Rayleigh-Bénard). Computer Methods in Applied Mechanics and Engineering, Vol. 261-262, pp. 132-141, 2013.

[2] Y. Maday, A.T. Patera and G. Turinici, Convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations. J. Sci. Comput., Vol. 7(1-4), pp. 437-446, 2002.

[3] F. Pla, A.M. Mancho and H. Herrero, Bifurcation phenomena in a convection problem with temperature dependent viscosity at low aspect ratio. Physica D, Vol 238, pp. 572-580, 2009.

[4] C. Prud’homme, D.V. Rovas, K. Veroy, L. Machiels, Y. Maday, A.T. Patera and G. Turinici, Reliable real-time solution of parametrized partial differential equations: Reducedbasis output bound methods, Journal of Fluids Engineering, Vol 124(1), pp. 70-80, 2002.

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Authors (presenter in bold)

Isabel Mercader Calvo, Universitat Politecnica de Catalunya
Oriol Batiste Boleda, Universitat Politecnica de Catalunya
Arantxa Alonso Maleta, Universitat Politecnica de Catalunya
Odalys Sanchez Casals, Universitat Politecnica de Catalunya

Abstract

We analyze here the flow in a horizontal closed cylinder of aspect ratio 2, induced by both an axial temperature gradient and the rotation about its axis. A Prandtl number representative of molten metals at high temperature is considered. We have characterized the symmetric basic states and analysed their stability when the temperature gradient increases in a region of moderate values of rotation rates, in which the two driving effects, natural convection and rotation, are comparable.

The basic state consists in a center-symmetric longitudinal flow, tilted with respect to the vertical plane and steady in the laboratory frame. Results characterizing the basic states as the rotation rate increases, in comparison with the non-rotating case previously studied (Mercader et al, 2014), are presented. As rotation increases, temperature becomes more uniform in space and the strength of the flow due to buoyancy in the vertical direction reduces. As it occurred in the non-rotating case for higher values of the Prandtl number, two curves of steady states with the same symmetric character coexist for moderate values of the Rayleigh number. In the range of values of rotation rate considered, rotation has a stabilizing effect only for very small values. As the rotation rate approaches to values of 3.5 and 4.5 in thermal units (length=diameter), the scenario of bifurcations becomes more complex due to the existence in both cases of very close bifurcations of codimension 2, which in the latter case involve both curves of symmetric solutions.

For the numerical calculations we use a second order time-splitting method for integrating the equations in time combined with a pseudo-spectral method for the spatial discretization (Mercader et al, 2010). Steady solutions and continuation of them are calculated by using the Stokes preconditioner developed by L Tuckermann (Mamun and Tuckerman, 1995)

The linear stability analysis has been carried on by using an Arnoldi method applied to an approximation of the Jacobian to obtain approximated leading eigenvalues and eigenvectors. To determine them accurately, the estimated eigenvalues and eigenvectors are used as initial guess to solve the nonlinear system derived from the eigenvalue problem.

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Authors (presenter in bold)

Damián Castaño Torrijos, UCLM
María Cruz Navarro Lérida, UCLM
Henar Herrero Sanz, UCLM

Abstract

In this talk, we study several routes of the transition to chaos from a steady axisymmetric vertical vortex in a rotating cylinder depending on thermal gradients and rotation rates. The analysis is done using nonlinear simulations. For a fixed rotation rate, the chaotic regime appears, as thermal gradients increase, after a sequence of supercritical Hopf bifurcations to periodic, quasiperiodic and chaotic flows in a scenario similar to the Ruelle - Takens - Newhouse route to chaos. For moderate values of the rotation rate we find vortices that tilt and move away from the center of the cylinder in a periodic, quasiperiodic and finally chaotic movement around the central axis. For larger rotation rates the axisymmetric vortex splits into two symmetric vortices that move periodically around the central axis, and lose the symmetry merging again in one non-axisymmetric vortex that moves around the central axis quasiperiodically and later chaotically. The transitions to chaos when the rotation rate is varied at fixed thermal gradients reveal also the appearance of periodic, quasiperiodic and chaotic states in different routes. Tilted single vortices, double vortices and more complex structures with multiple vortices are reported in this case. The transitions are studied through a force balance analysis. Results are of interest as they connect to the behavior of some atmospheric vertical vortices.

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Authors (presenter in bold)

sergio Gandía-Barberá, Universitat Politècnica de València
Sergio Hoyas, Universitat Politècnica de València
Jezabel Pérez-Quiles, Universitat Politècnica de València

Abstract

Turbulent flows are intrinsic to almost any flow in engineering, and their dynamic is probably the open problem in physics with most applications in day-by-day life. One of the characteristics of these flows that is getting a growing interest in the last years is the existence and stability of Large and Very Large Scale Motions (LSM and VLSM). One of the classical flows where this kind of structures appears is the Turbulent Couette FLOW (TCF). TCF shows extremely long and stable structures that are not present in any other wall bounded flow. There exists evidence of these structures in both numerical and experimental works, but little is known about their formation or destruction. The domain we have used is a turbulent channel wide enough to simulate several of these structures. Different channel lengths have been simulated to study the maximum length of the VLSM. Moreover, we have also computed several flows with mixed Poiseuille-Couette conditions, to understand the effect of pressure gradients on these structures.

The code used to solve the flow has been developed for more than twenty years. For the temporal discretization it uses a low storage third order Runge-Kutta scheme. The spatial directions are solved using a Fourier-Compact Finite Differences-Fourier method in the streamwise, normal and spanwise directions respectively. The meshes used are fine enough to capture the smallest scales of the flow. Statistics are computed for at least 100 wash-outs, i.e. the time needed for a typical eddy to travel through the channel. In this talk we will present the numerical method that we use for computing these structures, the algorithms developed for visualizing them, and the different strategies we have used to understand their formation and their stability.

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Authors (presenter in bold)

Ferran Garcia Gonzalez, Helmholtz-Zentrum Dresden-Rossendorf, University of Amsterdam
Juan Sánchez, Universitat Politècnica de Catalunya
Marta Net, Universitat Politècnica de Catalunya
Frank Chambers, University of Amsterdam
Anna Watts, University of Amsterdam
Frank Stefani, Helmholtz-Zentrum Dresden-Rossendorf

Abstract

Fluid dynamics plays an important role in many geophysical and astrophysical objects such as planets and stars. For instance, convection can occur in neutron stars' oceans formed by very thin layers of helium or hydrogen, which are subject to the influence of strong temperature gradients and rotation.

In addition, instabilities observed in differentially rotating flows in the presence of a magnetic field (magnetized spherical Couette flows) were attributed to the magnetorotational instability (MRI), which is presently considered the most promising candidate to explain the transport mechanism of angular momentum in accretion disks around black holes and protostars.

In this study, bifurcation diagrams of the first instabilities occurring in the two mentioned set-ups will be presented. They were obtained by means of continuation techniques. The arising flow patterns will be described. In both cases, pseudo-spectral high order methods as well as high order time integration methods are used for the time evolution of the Navier-Stokes equations.

Minisymposium Organisers

Xiong Meng, Harbin Institute of Technology
Jennifer K. Ryan, University of East Anglia
Qiang Zhang, Nanjing University

Abstract

Many aspects of superconvergence have been explored over the years and its relevance to accurate long-time wave propagation is only recently being appreciated, especially for numerical methods with discontinuous finite element spaces. The property of superconvergence comes in many forms -- pointwise, in the dispersion and dissipation error, special norms, and in a supercloseness property. It is an important phenomenon that not only results in accurate wave propagation, but also superconvergent recovery techniques for both the solution and its derivatives as well as adaptivity techniques. It is one of the important inherent features of discontinuous Galerkin methods that is often overlooked. This minisymposia has gathered researchers from all aspects of superconvergence to stimulate active discussion on the many aspects of superconvergence, how they are related, and how it can be exploited to design better numerical schemes.

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Authors (presenter in bold)

Qiang Zhang, Nanjing University

Abstract

In this talk we shall consider the local discontinuous Galerkin methods with the generalized upwind numerical flux and generalized alternating numerical fluxes, when solving the singularity perturbation problem with a stationary outflow boundary layer. The the double-optimal local L$^2$-norm error estimate is obtained, that is to say that, not only the width of pollution domain is optimal, and also the order of L$^2$-norm error out of that is optimal. To this purpose, the generalized Gauss-Radau (GGR) projection will play the important role. More difficult than the case in the global estimate, we have to establish a more deep investigation on the GGR projection and understand its global essence, by using the suitable weight functions nearby the outflow boundary. Time-marching techniques are considered also, such as the explicit second-order and third-order Runge-Kutta algorithm.

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Authors (presenter in bold)

Qi Tao, University of Science and Technology of China
Yinhua Xia, University of Science and Technology of China
Yan Xu, University of Science and Technology of China

Abstract

In this paper, we establish negative-order norm estimates for the accuracy of Arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) approximations to scalar nonlinear hyperbolic equations with smooth solutions. For these special solutions, we are able to extract this ``hidden accuracy'' through the use of a convolution kernel that is composed of a linear combination of B-splines. Previous investigations into extracting the superconvergence of DG methods using a convolution kernel have focused on fixed grid method. However, we now demonstrate that it is possible to extend the Smoothness-Increasing Accuracy-Conserving (SIAC) filter for ALE-DG methods on moving mesh. Furthermore, we provide theoretical error estimates of the ALE-DG solutions that show improvement to (2$k$+$m)-$th order in the negative-order norm, where $m$ depends upon the chosen flux.