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A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes


Abgrall, Rémi (2018). A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes. Journal of Computational Physics, 372:640-666.

Abstract

We are interested in the approximation of a steady hyperbolic problem. In some cases, the solution can satisfy an additional conservation relation, at least when it is smooth. This is the case of an entropy. In this paper, we show, starting from the discretization of the original PDE, how to construct a scheme that is consistent with the original PDE and the additional conservation relation. Since one interesting example is given by the systems endowed by an entropy, we provide one explicit solution, and show that the accuracy of the new scheme is at most degraded by one order. In the case of a discontinuous Galerkin scheme and a Residual distribution scheme, we show how not to degrade the accuracy. This improves the recent results obtained in [1], [2], [3], [4] in the sense that no particular constraints are set on quadrature formula and that a priori maximum accuracy can still be achieved. We study the behavior of the method on a non linear scalar problem. However, the method is not restricted to scalar problems.

Abstract

We are interested in the approximation of a steady hyperbolic problem. In some cases, the solution can satisfy an additional conservation relation, at least when it is smooth. This is the case of an entropy. In this paper, we show, starting from the discretization of the original PDE, how to construct a scheme that is consistent with the original PDE and the additional conservation relation. Since one interesting example is given by the systems endowed by an entropy, we provide one explicit solution, and show that the accuracy of the new scheme is at most degraded by one order. In the case of a discontinuous Galerkin scheme and a Residual distribution scheme, we show how not to degrade the accuracy. This improves the recent results obtained in [1], [2], [3], [4] in the sense that no particular constraints are set on quadrature formula and that a priori maximum accuracy can still be achieved. We study the behavior of the method on a non linear scalar problem. However, the method is not restricted to scalar problems.

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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 > Numerical Analysis
Physical Sciences > Modeling and Simulation
Physical Sciences > Physics and Astronomy (miscellaneous)
Physical Sciences > General Physics and Astronomy
Physical Sciences > Computer Science Applications
Physical Sciences > Computational Mathematics
Physical Sciences > Applied Mathematics
Uncontrolled Keywords:Physics and Astronomy (miscellaneous), Computer Science Applications
Language:English
Date:1 November 2018
Deposited On:01 Nov 2018 11:42
Last Modified:15 Apr 2020 21:43
Publisher:Elsevier
ISSN:0021-9991
OA Status:Closed
Publisher DOI:https://doi.org/10.1016/j.jcp.2018.06.031
Project Information:
  • : FunderSNSF
  • : Grant ID200021_153604
  • : Project TitleHigh fidelity simulation for compressible material

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