We discuss two approaches for the formulation and implementation of space-time discontinuous Galerkin spectral element methods (DG-SEM). In one, time is treated as an additional coordinate direction and a Galerkin procedure is applied to the entire problem. In the other, the method of lines is used with DG-SEM in space and the fully implicit Runge–Kutta method Lobatto IIIC in time. The two approaches are mathematically equivalent in the sense that they lead to the same discrete solution. However, in practice they differ in several important respects, including the terminology used to describe them, the structure of the resulting software, and the interaction with nonlinear solvers. Challenges and merits of the two approaches are discussed with the goal of providing the practitioner with sufficient consideration to choose which path to follow. Additionally, implementations of the two methods are provided as a starting point for further development. Numerical experiments validate the theoretical accuracy of these codes and demonstrate their utility, even for 4D problems.
Keywords: Space-time, Discontinuous Galerkin, DG-SEM
CC-BY 4.0
@article{SMAI-JCM_2023__9__61_0,
author = {Lea Miko Versbach and Viktor Linders and Robert Kl\"ofkorn and Philipp Birken},
title = {Theoretical and {Practical} {Aspects} of {Space-Time} {DG-SEM} {Implementations}},
journal = {The SMAI Journal of computational mathematics},
pages = {61--93},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {9},
year = {2023},
doi = {10.5802/smai-jcm.95},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.95/}
}
TY - JOUR AU - Lea Miko Versbach AU - Viktor Linders AU - Robert Klöfkorn AU - Philipp Birken TI - Theoretical and Practical Aspects of Space-Time DG-SEM Implementations JO - The SMAI Journal of computational mathematics PY - 2023 SP - 61 EP - 93 VL - 9 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.95/ DO - 10.5802/smai-jcm.95 LA - en ID - SMAI-JCM_2023__9__61_0 ER -
%0 Journal Article %A Lea Miko Versbach %A Viktor Linders %A Robert Klöfkorn %A Philipp Birken %T Theoretical and Practical Aspects of Space-Time DG-SEM Implementations %J The SMAI Journal of computational mathematics %D 2023 %P 61-93 %V 9 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.95/ %R 10.5802/smai-jcm.95 %G en %F SMAI-JCM_2023__9__61_0
Lea Miko Versbach; Viktor Linders; Robert Klöfkorn; Philipp Birken. Theoretical and Practical Aspects of Space-Time DG-SEM Implementations. The SMAI Journal of computational mathematics, Volume 9 (2023), pp. 61-93. doi : 10.5802/smai-jcm.95. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.95/
[1] Unified Form Language: A Domain-Specific Language for Weak Formulations of Partial Differential Equations, ACM Trans. Math. Softw., Volume 40 (2014) no. 2 | DOI | MR | Zbl
[2] Assimulo: A unified framework for ODE solvers, Math. Comput. Simul., Volume 116 (2015) no. 0, pp. 26-43 | DOI | MR | Zbl
[3] Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., Volume 39 (2002) no. 5, pp. 1749-1779 | DOI | MR | Zbl
[4] PETSc Web page, https://petsc.org/, 2021
[5] PETSc/TAO Users Manual (2021) no. ANL-21/39 - Revision 3.16 (Technical report)
[6] The Dune framework: Basic concepts and recent developments, Comput. Math. Appl., Volume 81 (2021), pp. 75-112 http://www.sciencedirect.com/science/article/pii/s089812212030256x | DOI | MR
[7] An efficient solution process for implicit Runge–Kutta methods, SIAM J. Numer. Anal., Volume 14 (1977) no. 6, pp. 1022-1027 | DOI | MR | Zbl
[8] Numerical Methods for Unsteady Compressible Flow Problems, CRC Press, 2021 | DOI
[9] Subcell finite volume multigrid preconditioning for high-order discontinuous Galerkin methods, Int. J. Comput. Fluid Dyn., Volume 33 (2019) no. 9, pp. 353-361 | DOI | MR | Zbl
[10] High-Order Implicit Time-Marching Methods Based on Generalized Summation-by-Parts Operators, SIAM J. Sci. Comput., Volume 37 (2015), p. A2682-A2709 | DOI | MR | Zbl
[11] Compact and stable Discontinuous Galerkin methods for convection-diffusion problems., SIAM J. Sci. Comput., Volume 34 (2012) no. 1, pp. 263-282 | DOI | MR | Zbl
[12] Implicit Runge-Kutta Processes, Math. Comput., Volume 18 (1964) no. 85, pp. 50-64 | DOI | MR | Zbl
[13] On the implementation of implicit Runge-Kutta methods, BIT, Volume 16 (1976) no. 3, pp. 237-240 | DOI | MR | Zbl
[14] Four-dimensional anisotropic mesh adaptation, Comput.-Aided Des., Volume 129 (2020), p. 102915 | DOI | MR
[15] Entropy stable spectral collocation schemes for the Navier–Stokes equations: Discontinuous interfaces, SIAM J. Sci. Comput., Volume 36 (2014) no. 5, p. B835-B867 | DOI | MR | Zbl
[16] Spectral methods on arbitrary grids, J. Comput. Phys., Volume 129 (1996) no. 1, pp. 74-86 | DOI | MR | Zbl
[17] A space-time discontinuous Galerkin method applied to single-phase flow in porous media, Comput. Geosci., Volume 12 (2008) no. 4, pp. 525-539 | DOI | MR | Zbl
[18] The DUNE-FEM-DG module, Archive of Numerical Software, Volume 5 (2017) no. 1 | DOI
[19] Extendible and Efficient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Method, Commun. Appl. Math. Comput. Sci. (2021) | DOI | Zbl
[20] The DUNE-FEM tutorial, 2022 (https://dune-project.org/sphinx/content/sphinx/dune-fem/)
[21] Python Bindings for the DUNE-FEM Module (2020) Zenodo (Mar 2020), https://doi.org/10.5281/zenodo.3706994
[22] A Generic Interface for Parallel and Adaptive Scientific Computing: Abstraction Principles and the DUNE-FEM Module, Computing, Volume 90 (2010) no. 3–4, pp. 165-196 | DOI | Zbl
[23] Construction of Local Finite Element Spaces Using the Generic Reference Elements, Advances in DUNE (A. Dedner; B. Flemisch; R. Klöfkorn, eds.), Springer (2012), pp. 3-16 | DOI
[24] Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations, Comput. Fluids, Volume 95 (2014), pp. 171-196 | DOI | MR | Zbl
[25] Tensor-product preconditioners for higher-order space–time discontinuous Galerkin methods, J. Comput. Phys., Volume 330 (2017), pp. 296-318 | DOI | MR | Zbl
[26] Space-time discontinuous Galerkin discretizations for linear first-order hyperbolic evolution systems, Comput. Methods Appl. Math., Volume 16 (2016) no. 3, pp. 409-428 | DOI | MR | Zbl
[27] High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains, J. Comput. Phys., Volume 252 (2013), pp. 518-557 | DOI | MR | Zbl
[28] Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: Theory and boundary conditions, J. Comput. Phys., Volume 234 (2013), pp. 353-375 | DOI | MR | Zbl
[29] Multigrid preconditioning for a space-time spectral-element discontinuous-Galerkin solver, AIAA Scitech 2020 Forum (2020), p. 1314 | DOI
[30] Local Fourier analysis of space-time relaxation and multigrid schemes, SIAM J. Sci. Comput., Volume 35 (2013) no. 5, p. S250-S276 | DOI | MR | Zbl
[31] Entropy Stable Space–Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws, J. Sci. Comput., Volume 80 (2019) no. 1, pp. 175-222 | DOI | MR | Zbl
[32] Foundations of space-time finite element methods: Polytopes, interpolation, and integration, Appl. Numer. Math., Volume 166 (2021), pp. 92-113 | DOI | MR | Zbl
[33] 50 Years of Time Parallel Time Integration, Multiple Shooting and Time Domain Decomposition Methods (T. Carraro; M. Geiger; S. Körkel; R. Rannacher, eds.), Springer (2015), pp. 69-113 | DOI | Zbl
[34] Analysis of a new space-time parallel multigrid algorithm for parabolic problems, SIAM J. Sci. Comput., Volume 38 (2016) no. 4, p. A2173-A2208 | DOI | MR | Zbl
[35] A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods, SIAM J. Sci. Comput., Volume 35 (2013) no. 3, p. A1233-A1253 | DOI | MR | Zbl
[36] A novel robust strategy for discontinuous Galerkin methods in computational fluid mechanics: Why? When? What? Where?, Front. Phys. (2021), p. 612
[37] Higher-order discontinuous finite element methods and dynamic model adaptation for hyperbolic systems of conservation laws, PhD thesis, University of Freiburg (2017) | DOI
[38] A multilevel discontinuous Galerkin method, Numer. Math., Volume 95 (2003) no. 3, pp. 527-550 | DOI | MR | Zbl
[39] Solving Ordinary Differential Equations I, Springer, 2009 | DOI
[40] Solving Ordinary Differential Equations II, Springer Series in Computational Mathematics, 14, Springer, 2010, xvi+614 pages | DOI
[41] Two-level Fourier Analysis of a Multigrid Approach for Discontinuous Galerkin Discretization, SIAM J. Sci. Comput., Volume 25 (2003) no. 3, pp. 1018-1041 | DOI | MR | Zbl
[42] Fourier two-level analysis for discontinuous Galerkin discretization with linear elements, Numer. Linear Algebra Appl., Volume 11 (2004) no. 5-6, pp. 473-491 | DOI | MR | Zbl
[43] Nodal discontinuous Galerkin methods: algorithms, analysis, and applications, Springer, 2008 | DOI
[44] Superconvergent functional estimates from summation-by-parts finite-difference discretizations, SIAM J. Sci. Comput., Volume 33 (2011) no. 2, pp. 893-922 | DOI | MR | Zbl
[45] SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers, ACM Trans. Math. Softw., Volume 31 (2005) no. 3, pp. 363-396 | DOI | MR | Zbl
[46] Automatic Symbolic Computation for Discontinuous Galerkin Finite Element Methods, SIAM J. Sci. Comput., Volume 40 (2018) no. 3, p. C327-C357 | DOI | MR | Zbl
[47] Space-time finite element methods for second-order hyperbolic equations, Comput. Methods Appl. Math., Volume 84 (1990) no. 3, pp. 327-348 | MR | Zbl
[48] Evaluation of fully implicit Runge Kutta schemes for unsteady flow calculations, J. Sci. Comput., Volume 73 (2017) no. 2-3, pp. 819-852 | DOI | MR | Zbl
[49] Lobatto Methods, Encyclopedia of Applied and Computational Mathematics (B. Engquist, ed.), Springer, 2015, pp. 817-826 | DOI
[50] A space-time adaptive method for reservoir flows: formulation and one-dimensional application, Comput. Geosci., Volume 22 (2018) no. 1, pp. 107-123 | DOI | MR | Zbl
[51] Optimal and Low-Memory Near-Optimal Preconditioning of Fully Implicit Runge–Kutta Schemes for Parabolic PDEs, SIAM J. Sci. Comput., Volume 43 (2021) no. 5, p. A3527-A3551 | DOI | MR | Zbl
[52] A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method, Math. Comput., Volume 67 (1998) no. 222, pp. 479-499 | DOI | Zbl
[53] Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford University Press, 2013 | DOI
[54] An Finite Volume Based Multigrid Preconditioner for DG-SEM for Convection-Diffusion, 14th WCCM-ECCOMAS Congress 2020, Volume 600 (2021) | DOI
[55] Space–time discontinuous Galerkin method for the compressible Navier–Stokes equations, J. Comput. Phys., Volume 217 (2006) no. 2, pp. 589-611 | DOI | MR | Zbl
[56] h-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier-Stokes equations, J. Comput. Phys., Volume 227 (2007) no. 2, pp. 1024-1045 | DOI | MR | Zbl
[57] Variational space–time methods for the wave equation, J. Sci. Comput., Volume 61 (2014) no. 2, pp. 424-453 | DOI | MR | Zbl
[58] Implementing Spectral Methods for Partial Differential Equations, Springer, 2009 | DOI
[59] On the Quadrature and Weak Form Choices in Collocation Type Discontinuous Galerkin Spectral Element Methods, J. Sci. Comput., Volume 44 (2010), pp. 136-155 | DOI | MR | Zbl
[60] Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method, Int. J. Numer. Methods Eng., Volume 53 (2002) no. 1, pp. 105-122 | DOI | Zbl
[61] et al. FLEXI: A high order discontinuous Galerkin framework for hyperbolic–parabolic conservation laws, Comput. Math. Appl., Volume 81 (2021), pp. 186-219 | DOI | MR | Zbl
[62] Finite Element and Finite Difference Methods for Hyperbolic Partial Differential Equations, Mathematical Aspects of Finite Elements in Partial Differential Equations (C. de Boor, ed.), Academic Press Inc., 1974, pp. 195-212 | DOI | Zbl
[63] A space-time finite element method for solving linear Riesz space fractional partial differential equations, Numer. Algorithms (2021), pp. 1-22
[64] Implementation of 3 stage Lobatto IIIC into Assimulo package, Bachelor thesis, Lund University (2021)
[65] On an eigenvalue property of Summation-By-Parts operators, J. Sci. Comput., Volume 93 (2022), p. 82 | DOI | MR | Zbl
[66] On the order of accuracy of finite difference operators on diagonal norm based summation-by-parts form, SIAM J. Numer. Anal., Volume 56 (2018) no. 2, pp. 1048-1063 | DOI | MR | Zbl
[67] Properties of Runge-Kutta-Summation-By-Parts methods, J. Comput. Phys., Volume 419 (2020), p. 109684 | DOI | MR | Zbl
[68] The SBP–SAT technique for initial value problems, J. Comput. Phys., Volume 270 (2014), pp. 86-104 | DOI | MR | Zbl
[69] A posteriori error analysis for higher order dissipative methods for evolution problems, Numer. Math., Volume 104 (2006) no. 4, pp. 489-514 | DOI | MR | Zbl
[70] A New Block Preconditioner for Implicit Runge–Kutta Methods for Parabolic PDE Problems, SIAM J. Sci. Comput., Volume 43 (2021) no. 5, p. S475-S495 | DOI | MR | Zbl
[71] Strong scaling for numerical weather prediction at petascale with the atmospheric model NUMA, Int. J. High Perform. Comput. Appl., Volume 33 (2019) no. 2, pp. 411-426 | DOI
[72] A High-Order Multiscale Global Atmospheric Model, AIAA AVIATION Forum (2016) | DOI
[73] Space-Time Methods, Monograph Series TU Graz: Computation in Engineering and Science, 20, TU Graz, 2013
[74] Parallel methods for integrating ordinary differential equations, Commun. ACM, Volume 7 (1964) no. 12, pp. 731-733 | DOI | MR | Zbl
[75] Energy stable boundary conditions for the nonlinear incompressible Navier–Stokes equations, Math. Comput., Volume 88 (2019) no. 316, pp. 665-690 | DOI | MR | Zbl
[76] Well-posed and stable transmission problems, J. Comput. Phys., Volume 364 (2018), pp. 95-110 | DOI | MR | Zbl
[77] Summation-by-parts in time, J. Comput. Phys., Volume 251 (2013), pp. 487-499 | DOI | MR | Zbl
[78] Stage-parallel fully implicit Runge–Kutta solvers for discontinuous Galerkin fluid simulations, J. Comput. Phys., Volume 335 (2017), pp. 700-717 | DOI | MR | Zbl
[79] Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods, J. Comput. Phys., Volume 354 (2018), pp. 344-369 | DOI | MR | Zbl
[80] Some notes on summation by parts time integration methods, Results Appl. Math., Volume 1 (2019), p. 100004 | DOI | MR | Zbl
[81] Adaptive numerical simulations with Trixi.jl: A case study of Julia for scientific computing, Proceedings of the JuliaCon Conferences, Volume 1 (2022) no. 1, p. 77 | DOI
[82] On pseudo-spectral time discretizations in summation-by-parts form, J. Comput. Phys., Volume 360 (2018), pp. 192-201 | DOI | MR | Zbl
[83] B-convergence of Lobatto IIIC formulas, Numer. Math., Volume 51 (1987) no. 2, pp. 229-235 | DOI | MR | Zbl
[84] High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments, J. Comput. Phys., Volume 316 (2016), pp. 598-613 | DOI | MR | Zbl
[85] Summation by parts for finite difference approximations for d/dx, J. Comput. Phys., Volume 110 (1994) no. 1, pp. 47-67 | DOI | MR | Zbl
[86] Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependent domains, Appl. Numer. Math., Volume 56 (2006) no. 12, pp. 1491-1518 | DOI | MR | Zbl
[87] Review of summation-by-parts schemes for initial–boundary-value problems, J. Comput. Phys., Volume 268 (2014), pp. 17-38 | DOI | MR | Zbl
[88] Space–time computations in practical engineering applications: A summary of the 25-year history, Comput. Mech., Volume 63 (2019) no. 4, pp. 747-753 | DOI | Zbl
[89] Space-time discontinuous Galerkin finite element methods, Von Karman Institute for Fluid Dynamics (2006), pp. 1-37
[90] hp-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows: Part I. Multilevel analysis, J. Comput. Phys., Volume 231 (2012) no. 22, pp. 7537-7563 | DOI | MR | Zbl
[91] hp-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II: Optimization of the Runge–Kutta smoother, J. Comput. Phys., Volume 231 (2012), pp. 7564-7583 | DOI | MR | Zbl
[92] Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows. I. General formulation, J. Comput. Phys., Volume 182 (2002) no. 2, pp. 546-585 | DOI | MR | Zbl
[93] Space–time discontinuous Galerkin method for nonlinear water waves, J. Comput. Phys., Volume 224 (2007) no. 1, pp. 17-39 | DOI | MR | Zbl
[94] Space–time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows: II. Efficient flux quadrature, Comput. Methods Appl. Math., Volume 191 (2002) no. 41-42, pp. 4747-4780 | DOI | MR | Zbl
[95] Two-level multigrid analysis for the convection–diffusion equation discretized by a discontinuous Galerkin method, Numer. Linear Algebra Appl., Volume 12 (2005) no. 5-6, pp. 563-584 | DOI | MR | Zbl
[96] Efficient Solvers for Space-Time Discontinuous Galerkin Spectral Element Methods, Ph. D. Thesis, Mathematics (Faculty of Sciences), Lund University (2022)
[97] Local Fourier Analysis of a Space-Time Multigrid Method for DG-SEM for the Linear Advection Equation (2021) (https://arxiv.org/abs/2112.03115)
[98] A third-order implicit discontinuous Galerkin method based on a Hermite WENO reconstruction for time-accurate solution of the compressible Navier–Stokes equations, Int. J. Numer. Methods Fluids, Volume 79 (2015) no. 8, pp. 416-435 | DOI | MR | Zbl
[99] An optimization-based framework for anisotropic simplex mesh adaptation, J. Comput. Phys., Volume 231 (2012) no. 22, pp. 7626-7649 | DOI | MR | Zbl
[100] Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws, SIAM J. Numer. Anal., Volume 44 (2006) no. 4, pp. 1703-1720 | DOI | MR | Zbl
Cited by Sources: