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Relaxation deferred correction methods and their applications to residual distribution schemes


Abgrall, Rémi; Le Mélédo, Elise; Öffner, Philipp; Torlo, Davide (2022). Relaxation deferred correction methods and their applications to residual distribution schemes. SMAI journal of computational mathematics, 8:125-160.

Abstract

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.

Abstract

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Statistics and Probability
Physical Sciences > Numerical Analysis
Physical Sciences > Modeling and Simulation
Physical Sciences > Computational Mathematics
Uncontrolled Keywords:Computational Mathematics, Modeling and Simulation, Numerical Analysis, Statistics and Probability elaxation, entropy conservative / dissipation, deferred correction, residual distribution.
Language:English
Date:13 October 2022
Deposited On:12 Dec 2022 14:55
Last Modified:22 Jun 2024 03:34
Publisher:Société de mathématiques appliquées et industrielles
ISSN:2426-8399
Additional Information:2020 Mathematics Subject Classification. 65M60, 65L05
OA Status:Gold
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.5802/smai-jcm.82
  • Content: Published Version
  • Language: English
  • Licence: Creative Commons: Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)