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Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes
Rémi Abgrall1; Élise Le Mélédo1; Philipp Öffner2; Davide Torlo3
1 Institute of Mathematics,University of Zurich, Switzerland
2 Institute of Mathematics, Johannes Gutenberg-University Mainz, Germany
3 SISSA mathLab, SISSA, Trieste, Italy
The SMAI Journal of computational mathematics, Volume 8 (2022), pp. 125-160.

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.

Published online:
DOI: 10.5802/smai-jcm.82
Classification: 65M60, 65L05
Keywords: relaxation, entropy conservative / dissipation, deferred correction, residual distribution

Rémi Abgrall 1; Élise Le Mélédo 1; Philipp Öffner 2; Davide Torlo 3

1 Institute of Mathematics,University of Zurich, Switzerland
2 Institute of Mathematics, Johannes Gutenberg-University Mainz, Germany
3 SISSA mathLab, SISSA, Trieste, Italy
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {Relaxation {Deferred} {Correction} {Methods} and their {Applications} to {Residual} {Distribution} {Schemes}},
     journal = {The SMAI Journal of computational mathematics},
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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/

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