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Theoretical and Practical Aspects of Space-Time DG-SEM Implementations
Lea Miko Versbach1; Viktor Linders1; Robert Klöfkorn1; Philipp Birken1
1 Centre for Mathematical Sciences, Numerical Analysis, Lund University, Lund, Sweden
The SMAI Journal of computational mathematics, Volume 9 (2023), pp. 61-93.

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.

Published online:
DOI: 10.5802/smai-jcm.95
Classification: 65M60, 65M99
Mots-clés : Space-time, Discontinuous Galerkin, DG-SEM

Lea Miko Versbach 1; Viktor Linders 1; Robert Klöfkorn 1; Philipp Birken 1

1 Centre for Mathematical Sciences, Numerical Analysis, Lund University, Lund, Sweden
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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] M. S. Alnæs; A. Logg; K. B. Ølgaard; M. E. Rognes; G. N. Wells 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] C. Andersson; C. Führer; J. Åkesson Assimulo: A unified framework for ODE solvers, Math. Comput. Simul., Volume 116 (2015) no. 0, pp. 26-43 | DOI | MR | Zbl

[3] D. N. Arnold; F. Brezzi; B. Cockburn; L. D. Marini 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] S. Balay; S. Abhyankar; M. F. Adams; S. Benson; J. Brown; P. Brune; K. Buschelman; E. M. Constantinescu; L. Dalcin; A. Dener; V. Eijkhout; W. D. Gropp; V. Hapla; T. Isaac; P. Jolivet; D. Karpeev; D. Kaushik; M. G. Knepley; F. Kong; S. Kruger; D. A. May; L. C. McInnes; R. T. Mills; L. Mitchell; T. Munson; J. E. Roman; K. Rupp; P. Sanan; J. Sarich; B. F. Smith; S. Zampini; H. Zhang; H. Zhang; J. Zhang PETSc Web page, https://petsc.org/, 2021

[5] S. Balay; S. Abhyankar; M. F. Adams; S. Benson; J. Brown; P. Brune; K. Buschelman; E. M. Constantinescu; L. Dalcin; A. Dener; V. Eijkhout; W. D. Gropp; V. Hapla; T. Isaac; P. Jolivet; D. Karpeev; D. Kaushik; M. G. Knepley; F. Kong; S. Kruger; D. A. May; L. C. McInnes; R. T. Mills; L. Mitchell; T. Munson; J. E. Roman; K. Rupp; P. Sanan; J. Sarich; B. F. Smith; S. Zampini; H. Zhang; H. Zhang; J. Zhang PETSc/TAO Users Manual (2021) no. ANL-21/39 - Revision 3.16 (Technical report)

[6] P. Bastian; M. Blatt; M. Dedner; N.-A. Dreier; R. Engwer; C. Gräser; Ch. Grüninger; D. Kempf; R. Klöfkorn; M. Ohlberger; O. Sander 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] T. A. Bickart 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] P. Birken Numerical Methods for Unsteady Compressible Flow Problems, CRC Press, 2021 | DOI

[9] P. Birken; G. J. Gassner; L. M. Versbach 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] P. D. Boom; D. W. Zingg 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] S. Brdar; A. Dedner; R. Klöfkorn 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] J. C. Butcher Implicit Runge-Kutta Processes, Math. Comput., Volume 18 (1964) no. 85, pp. 50-64 | DOI | MR | Zbl

[13] J. C. Butcher On the implementation of implicit Runge-Kutta methods, BIT, Volume 16 (1976) no. 3, pp. 237-240 | DOI | MR | Zbl

[14] P. C. Caplan; R. Haimes; D. L. Darmofal; M. C. Galbraith Four-dimensional anisotropic mesh adaptation, Comput.-Aided Des., Volume 129 (2020), p. 102915 | DOI | MR

[15] M. H. Carpenter; T. C. Fisher; E. J. Nielsen; S. H. Frankel 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] M. H. Carpenter; D. Gottlieb Spectral methods on arbitrary grids, J. Comput. Phys., Volume 129 (1996) no. 1, pp. 74-86 | DOI | MR | Zbl

[17] Z. Chen; H. Steeb; S. Diebels 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] A. Dedner; S. Girke; R. Klöfkorn; T. Malkmus The DUNE-FEM-DG module, Archive of Numerical Software, Volume 5 (2017) no. 1 | DOI

[19] A. Dedner; R. Klöfkorn Extendible and Efficient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Method, Commun. Appl. Math. Comput. Sci. (2021) | DOI | Zbl

[20] A. Dedner; R. Klöfkorn The DUNE-FEM tutorial, 2022 (https://dune-project.org/sphinx/content/sphinx/dune-fem/)

[21] A. Dedner; R. Klöfkorn; M. Nolte Python Bindings for the DUNE-FEM Module (2020) Zenodo (Mar 2020), https://doi.org/10.5281/zenodo.3706994

[22] A. Dedner; R. Klöfkorn; M. Nolte; M. Ohlberger 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] A. Dedner; M. Nolte 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] D. C. Del Rey Fernández; J. E. Hicken; D. W. Zingg 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] L. T. Diosady; S. M. Murman Tensor-product preconditioners for higher-order space–time discontinuous Galerkin methods, J. Comput. Phys., Volume 330 (2017), pp. 296-318 | DOI | MR | Zbl

[26] W. Dörfler; S. Findeisen; C. Wieners 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] T. C. Fisher; M. H. Carpenter 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] T. C. Fisher; M. H. Carpenter; J. Nordström; N. K. Yamaleev; C. Swanson 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] M. Franciolini; S. M. Murman Multigrid preconditioning for a space-time spectral-element discontinuous-Galerkin solver, AIAA Scitech 2020 Forum (2020), p. 1314 | DOI

[30] S. Friedhoff; S. MacLachlan; C. Borgers 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] L. Friedrich; G. Schnücke; A. R. Winters; D. C. Del Rey Fernández; G. J. Gassner; M. H. Carpenter 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] C. V. Frontin; G. S. Walters; F. D. Witherden; C. W. Lee; D. M. Williams; D. L. Darmofal Foundations of space-time finite element methods: Polytopes, interpolation, and integration, Appl. Numer. Math., Volume 166 (2021), pp. 92-113 | DOI | MR | Zbl

[33] M. J. Gander 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] M. J. Gander; M. Neumüller 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] G. J. Gassner 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] G. J. Gassner; A. R. Winters A novel robust strategy for discontinuous Galerkin methods in computational fluid mechanics: Why? When? What? Where?, Front. Phys. (2021), p. 612

[37] Ch. Gersbacher Higher-order discontinuous finite element methods and dynamic model adaptation for hyperbolic systems of conservation laws, PhD thesis, University of Freiburg (2017) | DOI

[38] J. Gopalakrishnan; G. Kanschat A multilevel discontinuous Galerkin method, Numer. Math., Volume 95 (2003) no. 3, pp. 527-550 | DOI | MR | Zbl

[39] E. Hairer; S. P. Nørsett; G. Wanner Solving Ordinary Differential Equations I, Springer, 2009 | DOI

[40] E. Hairer; G. Wanner Solving Ordinary Differential Equations II, Springer Series in Computational Mathematics, 14, Springer, 2010, xvi+614 pages | DOI

[41] P. W. Hemker; W. Hoffmann; M. H. Van Raalte 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] P. W. Hemker; W. Hoffmann; M. H. Van Raalte 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] J. S. Hesthaven; T. Warburton Nodal discontinuous Galerkin methods: algorithms, analysis, and applications, Springer, 2008 | DOI

[44] J. E. Hicken; D. W. Zingg 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] A. C. Hindmarsh; P. N. Brown; K. E. Grant; S. L. Lee; R. Serban; D. E. Shumaker; C. S. Woodward 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] P. Houston; N. Sime 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] G. M. Hulbert; T. Hughes 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] A. Jameson 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] L. O. Jay Lobatto Methods, Encyclopedia of Applied and Computational Mathematics (B. Engquist, ed.), Springer, 2015, pp. 817-826 | DOI

[50] S. Jayasinghe; D. L. Darmofal; N. K. Burgess; M. C. Galbraith; S. R. Allmaras 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] X. Jiao; X. Wang; O. Chen 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] O. Karakashian; C. Makridakis 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] G. Karniadakis; S. Sherwin Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford University Press, 2013 | DOI

[54] J. Kasimir; L. M. Versbach; P. Birken; G. J. Gassner; R. Klöfkorn An Finite Volume Based Multigrid Preconditioner for DG-SEM for Convection-Diffusion, 14th WCCM-ECCOMAS Congress 2020, Volume 600 (2021) | DOI

[55] C. M. Klaij; J. J. W. van der Vegt; H. van der Ven 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] C. M. Klaij; M. H. van Raalte; H. van der Ven; J. J. W. van der Vegt 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] U. Köcher; M. Bause Variational space–time methods for the wave equation, J. Sci. Comput., Volume 61 (2014) no. 2, pp. 424-453 | DOI | MR | Zbl

[58] D. A. Kopriva Implementing Spectral Methods for Partial Differential Equations, Springer, 2009 | DOI

[59] D. A. Kopriva; G. J. Gassner 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] D. A. Kopriva; S. L. Woodruff; M. Y. Hussaini 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] N. Krais; A. Beck; T. Bolemann; H. Frank; D. Flad; G. J. Gassner; F. Hindenlang; M. Hoffmann; T. Kuhn; M. Sonntag 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] H.-O. Kreiss; G. Scherer 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] J. Lai; F. Liu; V. V. Anh; Q. Liu A space-time finite element method for solving linear Riesz space fractional partial differential equations, Numer. Algorithms (2021), pp. 1-22

[64] E. Lehsten Implementation of 3 stage Lobatto IIIC into Assimulo package, Bachelor thesis, Lund University (2021)

[65] V. Linders On an eigenvalue property of Summation-By-Parts operators, J. Sci. Comput., Volume 93 (2022), p. 82 | DOI | MR | Zbl

[66] V. Linders; T. Lundquist; J. Nordström 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] V. Linders; J. Nordström; S. H. Frankel Properties of Runge-Kutta-Summation-By-Parts methods, J. Comput. Phys., Volume 419 (2020), p. 109684 | DOI | MR | Zbl

[68] T. Lundquist; J. Nordström The SBP–SAT technique for initial value problems, J. Comput. Phys., Volume 270 (2014), pp. 86-104 | DOI | MR | Zbl

[69] C. Makridakis; R. H. Nochetto 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] M. Masud Rana; V. E. Howle; K. Long; A. Meek; W. Milestone 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] A. Müller; M. A. Kopera; S. Marras; L. C. Wilcox; T. Isaac; F. X. Giraldo 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] R. D. Nair; L. Bao; M. D. Toy; R. Klöfkorn A High-Order Multiscale Global Atmospheric Model, AIAA AVIATION Forum (2016) | DOI

[73] M. Neumüller Space-Time Methods, Monograph Series TU Graz: Computation in Engineering and Science, 20, TU Graz, 2013

[74] J. Nievergelt Parallel methods for integrating ordinary differential equations, Commun. ACM, Volume 7 (1964) no. 12, pp. 731-733 | DOI | MR | Zbl

[75] J. Nordström; C. La Cognata 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] J. Nordström; V. Linders Well-posed and stable transmission problems, J. Comput. Phys., Volume 364 (2018), pp. 95-110 | DOI | MR | Zbl

[77] J. Nordström; T. Lundquist Summation-by-parts in time, J. Comput. Phys., Volume 251 (2013), pp. 487-499 | DOI | MR | Zbl

[78] W. Pazner; P.-O. Persson 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] W. Pazner; P.-O. Persson Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods, J. Comput. Phys., Volume 354 (2018), pp. 344-369 | DOI | MR | Zbl

[80] H. Ranocha Some notes on summation by parts time integration methods, Results Appl. Math., Volume 1 (2019), p. 100004 | DOI | MR | Zbl

[81] H. Ranocha; M. Schlottke-Lakemper; A. R. Winters; E. Faulhaber; J. Chan; G. J. Gassner 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] A. A. Ruggiu; J. Nordström On pseudo-spectral time discretizations in summation-by-parts form, J. Comput. Phys., Volume 360 (2018), pp. 192-201 | DOI | MR | Zbl

[83] J. Schneid B-convergence of Lobatto IIIC formulas, Numer. Math., Volume 51 (1987) no. 2, pp. 229-235 | DOI | MR | Zbl

[84] C.-W. Shu 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] B. Strand 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] J. J. Sudirham; J. J. W. van der Vegt; R. M. J. van Damme 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] M. Svärd; J. Nordström Review of summation-by-parts schemes for initial–boundary-value problems, J. Comput. Phys., Volume 268 (2014), pp. 17-38 | DOI | MR | Zbl

[88] T. E. Tezduyar; K. Takizawa 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] J. J. W. van der Vegt Space-time discontinuous Galerkin finite element methods, Von Karman Institute for Fluid Dynamics (2006), pp. 1-37

[90] J. J. W. van der Vegt; S. Rhebergen 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] J. J. W. van der Vegt; S. Rhebergen 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] J. J. W. van der Vegt; H. van der Ven 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] J. J. W. van der Vegt; Y. Xu Space–time discontinuous Galerkin method for nonlinear water waves, J. Comput. Phys., Volume 224 (2007) no. 1, pp. 17-39 | DOI | MR | Zbl

[94] H. Van der Ven; J. J. W. van der Vegt 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] M. H. Van Raalte; P. W. Hemker 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] L. M. Versbach Efficient Solvers for Space-Time Discontinuous Galerkin Spectral Element Methods, Ph. D. Thesis, Mathematics (Faculty of Sciences), Lund University (2022)

[97] L. M. Versbach; P. Birken; V. Linders; G. J. Gassner 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] Y. Xia; X. Liu; H. Luo; R. Nourgaliev 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] M. Yano; D. L. Darmofal An optimization-based framework for anisotropic simplex mesh adaptation, J. Comput. Phys., Volume 231 (2012) no. 22, pp. 7626-7649 | DOI | MR | Zbl

[100] Q. Zhang; C.-W. Shu 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

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