The Deferred Correction (DeC) methods combined with the residual distribution (RD) approach allow the construction of high order continuous Galerkin (cG) schemes avoiding the inversion of the mass matrix. With the application of entropy correction functions we can even obtain entropy conservative/dissipative spatial discretizations in this context. To handle entropy production in time, a relaxation approach has been suggested by Ketcheson. The main idea is to slightly modify the time-step size such that the approximated solution fulfills the underlying entropy conservation/dissipation constraint. In this paper, we first study the relaxation technique applied to the DeC approach as an ODE solver, then we extend this combination to the residual distribution method, requiring more technical steps. The outcome is a class of cG methods that is fully entropy conservative/dissipative and where we can still avoid the inversion of a mass matrix.
Keywords: relaxation, entropy conservative / dissipation, deferred correction, residual distribution
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@article{SMAI-JCM_2022__8__125_0,
author = {R\'emi Abgrall and \'Elise Le M\'el\'edo and Philipp \"Offner and Davide Torlo},
title = {Relaxation {Deferred} {Correction} {Methods} and their {Applications} to {Residual} {Distribution} {Schemes}},
journal = {The SMAI Journal of computational mathematics},
pages = {125--160},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {8},
year = {2022},
doi = {10.5802/smai-jcm.82},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.82/}
}
TY - JOUR AU - Rémi Abgrall AU - Élise Le Mélédo AU - Philipp Öffner AU - Davide Torlo TI - Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes JO - The SMAI Journal of computational mathematics PY - 2022 SP - 125 EP - 160 VL - 8 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.82/ DO - 10.5802/smai-jcm.82 LA - en ID - SMAI-JCM_2022__8__125_0 ER -
%0 Journal Article %A Rémi Abgrall %A Élise Le Mélédo %A Philipp Öffner %A Davide Torlo %T Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes %J The SMAI Journal of computational mathematics %D 2022 %P 125-160 %V 8 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.82/ %R 10.5802/smai-jcm.82 %G en %F SMAI-JCM_2022__8__125_0
Rémi Abgrall; Élise Le Mélédo; Philipp Öffner; Davide Torlo. Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes. The SMAI Journal of computational mathematics, Volume 8 (2022), pp. 125-160. doi : 10.5802/smai-jcm.82. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.82/
[1] A review of residual distribution schemes for hyperbolic and parabolic problems: the July 2010 state of the art, Commun. Comput. Phys., Volume 11 (2012) no. 4, pp. 1043-1080 | DOI | MR | Zbl
[2] High Order Schemes for Hyperbolic Problems Using Globally Continuous Approximation and Avoiding Mass Matrices, J. Sci. Comput., Volume 73 (2017) no. 2, pp. 461-494 | DOI | MR | Zbl
[3] A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes, J. Comput. Phys., Volume 372 (2018), pp. 640-666 | DOI | MR | Zbl
[4] High-order residual distribution scheme for the time-dependent Euler equations of fluid dynamics, Comput. Math. Appl., Volume 78 (2019) no. 2, pp. 274-297 | DOI | MR | Zbl
[5] On the Connection between Residual Distribution Schemes and Flux Reconstruction (2018) (https://arxiv.org/abs/1807.01261)
[6] Error boundedness of correction procedure via reconstruction/flux reconstruction and the connection to residual distribution schemes, Hyperbolic Problems: Theory, Numerics, Applications (2018), p. 215 | Zbl
[7] Analysis of the SBP-SAT stabilization for finite element methods part I: Linear problems, J. Sci. Comput., Volume 85 (2020) no. 2, pp. 1-29 | MR | Zbl
[8] Analysis of the SBP-SAT stabilization for finite element methods part II: Entropy stability, Commun. Appl. Math. Comput. (2021), pp. 1-23
[9] Reinterpretation and Extension of Entropy Correction Terms for Residual Distribution and Discontinuous Galerkin Schemes: Application to Structure Preserving Discretization, J. Comput. Phys. (2022), p. 110955 | DOI | MR | Zbl
[10] High order asymptotic preserving deferred correction implicit-explicit schemes for kinetic models, SIAM J. Sci. Comput., Volume 42 (2020) no. 3, p. B816-B845 | DOI | MR | Zbl
[11] “A Posteriori” Limited High Order and Robust Residual Distribution Schemes for Transient Simulations of Fluid Flows in Gas Dynamics (2019) (https://arxiv.org/abs/1902.07773)
[12] Continuous interior penalty -finite element methods for advection and advection-diffusion equations, Math. Comput., Volume 76 (2007) no. 259, pp. 1119-1140 | DOI | MR | Zbl
[13] The edge stabilization method for finite elements in CFD, Numerical mathematics and advanced applications, Springer, 2004, pp. 196-203 | DOI
[14] Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws, J. Comput. Phys., Volume 345 (2017), pp. 427-461 | DOI | MR | Zbl
[15] Integral deferred correction methods constructed with high order Runge-Kutta integrators, Math. Comput., Volume 79 (2010) no. 270, pp. 761-783 | DOI | MR | Zbl
[16] A spatial high-order hexahedral discontinuous Galerkin method to solve Maxwell’s equations in time domain, J. Comput. Phys., Volume 217 (2006) no. 2, pp. 340-363 | DOI | MR
[17] Higher Order Triangular Finite Elements with Mass Lumping for the Wave Equation, SIAM J. Numer. Anal., Volume 38 (2001) | DOI | MR
[18] Interior Penalty Procedures for Elliptic and Parabolic Galerkin Method, 58, Springer (2008), pp. 207-216 | DOI
[19] Spectral deferred correction methods for ordinary differential equations, BIT Numer. Math., Volume 40 (2000) no. 2, pp. 241-266 | DOI | MR | Zbl
[20] Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points, Appl. Numer. Math., Volume 151 (2020), pp. 98-118 | DOI | MR | Zbl
[21] Artificial viscosity for correction procedure via reconstruction using summation-by-parts operators, XVI International Conference on Hyperbolic Problems: Theory, Numerics, Applications, Springer (2016), pp. 363-375 | Zbl
[22] High order strong stability preserving time discretizations, J. Sci. Comput., Volume 38 (2009) no. 3, pp. 251-289 | DOI | MR | Zbl
[23] DeC and ADER: Similarities, Differences and a Unified Framework, J. Sci. Comput., Volume 87 (2021) no. 1, pp. 1-35 | MR | Zbl
[24] On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys., Volume 49 (1983), pp. 151-164 | DOI | MR | Zbl
[25] Positivity-Preserving Time Discretizations for Production–Destruction Equations with Applications to Non-equilibrium Flows, J. Sci. Comput., Volume 78 (2019) no. 3, pp. 1811-1839 | DOI | MR | Zbl
[26] Relaxation Runge–Kutta Methods: Conservation and Stability for Inner-Product Norms, SIAM J. Numer. Anal., Volume 57 (2019) no. 6, pp. 2850-2870 | DOI | MR | Zbl
[27] A comparison of high-order explicit Runge–Kutta, extrapolation, and deferred correction methods in serial and parallel, Commun. Appl. Math. Comput. Sci., Volume 9 (2014) no. 2, pp. 175-200 | DOI | MR | Zbl
[28] Bound-preserving Flux Limiting for High-Order Explicit Runge–Kutta Time Discretizations of Hyperbolic Conservation Laws, J. Sci. Comput., Volume 91 (2022) no. 1, p. 21 | DOI | MR | Zbl
[29] Strong stability preserving property of the deferred correction time discretization, J. Comput. Math. (2008), pp. 633-656 | MR | Zbl
[30] On unconditionally positive implicit time integration for the DG scheme applied to shallow water flows, Int. J. Numer. Methods Fluids, Volume 76 (2014) no. 2, pp. 69-94 | DOI | MR | Zbl
[31] Spectral Analysis of Continuous FEM for Hyperbolic PDEs: Influence of Approximation, Stabilization, and Time-Stepping, J. Sci. Comput., Volume 89 (2021) no. 31 | MR | Zbl
[32] Semi-implicit spectral deferred correction methods for ordinary differential equations, Commun. Math. Sci., Volume 1 (2003) no. 3, pp. 471-500 | DOI | MR | Zbl
[33] Positivity-preserving adaptive Runge–Kutta methods, Commun. Appl. Math. Comput. Sci., Volume 16 (2021) no. 2, pp. 155-179 | DOI | MR | Zbl
[34] Approximation and Stability properties of Numerical Methods for Hyperbolic Conservation Laws, Habilitation Thesis, University Zurich (2020)
[35] Analysis of Artificial Dissipation of Explicit and Implicit Time-Integration Methods, Int. J. Numer. Anal. Model., Volume 17 (2020) no. 3, pp. 332-349 | MR | Zbl
[36] Arbitrary high-order, conservative and positivity preserving Patankar-type deferred correction schemes, Appl. Numer. Math., Volume 153 (2020), pp. 15-34 | DOI | MR | Zbl
[37] Relaxation Runge–Kutta Methods for inner-product norms, https://github.com/ketch/RRK_rr, 2019
[38] Relaxation Runge-Kutta Methods for Hamiltonian Problems, J. Sci. Comput., Volume 84 (2020) no. 1, pp. 1-27 | MR | Zbl
[39] General relaxation methods for initial-value problems with application to multistep schemes, Numer. Math., Volume 146 (2020) no. 4, pp. 875-906 | DOI | MR | Zbl
[40] Summation-by-parts operators for correction procedure via reconstruction, J. Comput. Phys., Volume 311 (2016), pp. 299-328 | arXiv | DOI | MR | Zbl
[41] Relaxation Runge–Kutta Methods: Fully Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier–Stokes Equations, SIAM J. Sci. Comput., Volume 42 (2020) no. 2, p. A612-A638 | DOI | MR | Zbl
[42] Explicit Runge-Kutta residual distribution schemes for time dependent problems: second order case, J. Comput. Phys., Volume 229 (2010) no. 16, pp. 5653-5691 | DOI | MR | Zbl
[43] Analytical travelling vortex solutions of hyperbolic equations for validating very high order schemes (2021) (https://arxiv.org/abs/2109.10183)
[44] Hyperbolic Problems: High Order Methods and Model Order Reduction, Ph. D. Thesis, University Zurich (2020)
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