<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>We present an entropy stable Discontinuous Galerkin (DG) finite element method to approximate systems of 2-dimensional symmetrizable conservation laws on unstructured grids. The scheme is constructed using a combination of entropy conservative fluxes and entropy-stable numerical dissipation operators. The method is designed to work on structured as well as on unstructured meshes. As solutions of hyperbolic conservation laws can develop discontinuities (shocks) in finite time, we include a multidimensional slope limitation step to suppress spurious oscillations in the vicinity of shocks. The numerical scheme has two steps: the first step is a finite element calculation which includes calculations of fluxes across the edges of the elements using 1-D entropy stable solver. The second step is a procedure of stabilization through a truly multi-dimensional slope limiter. We compared the Entropy Stable Scheme (ESS) versus Roe’s solvers associated with entropy corrections and Osher’s solver. The method is illustrated by computing solution of the two stationary problems: a regular shock reflection problem and a 2-D flow around a double ellipse at high Mach number.</p>
</abstract>

We present an entropy stable Discontinuous Galerkin (DG) finite element method to approximate systems of 2-dimensional symmetrizable conservation laws on unstructured grids. The scheme is constructed using a combination of entropy conservative fluxes and entropy-stable numerical dissipation operators. The method is designed to work on structured as well as on unstructured meshes. As solutions of hyperbolic conservation laws can develop discontinuities (shocks) in finite time, we include a multidimensional slope limitation step to suppress spurious oscillations in the vicinity of shocks. The numerical scheme has two steps: the first step is a finite element calculation which includes calculations of fluxes across the edges of the elements using 1-D entropy stable solver. The second step is a procedure of stabilization through a truly multi-dimensional slope limiter. We compared the Entropy Stable Scheme (ESS) versus Roe’s solvers associated with entropy corrections and Osher’s solver. The method is illustrated by computing solution of the two stationary problems: a regular shock reflection problem and a 2-D flow around a double ellipse at high Mach number.

Entropy Stable Discontinuous Galerkin Finite Element Method with Multi-Dimensional Slope Limitation for Euler Equations

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

We present an entropy stable Discontinuous Galerkin (DG) finite element method to approximate systems of 2-dimensional symmetrizable conservation laws on...

Entropy Stable Discontinuous Galerkin Finite Element Method with Multi-Dimensional Slope Limitation for Euler Equations

Published Online: Jan 13, 2022

Page range: 102 - 127

Received: Mar 07, 2021

Accepted: Aug 12, 2021

DOI: https://doi.org/10.2478/mjpaa-2022-0009

Keywords
Discontinuous finite elements, Multi-dimensional slope limiter, Entropy Stable Scheme (ESS), Entropy Conservative Scheme (ECS), Entropy corrections, Unstructured grids

© 2021 Aziz Madrane et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Entropy Stable Discontinuous Galerkin Finite Element Method with Multi-Dimensional Slope Limitation for Euler Equations

Published Online: Jan 13, 2022

Page range: 102 - 127

Received: Mar 07, 2021

Accepted: Aug 12, 2021

DOI: https://doi.org/10.2478/mjpaa-2022-0009

KeywordsDiscontinuous finite elements, Multi-dimensional slope limiter, Entropy Stable Scheme (ESS), Entropy Conservative Scheme (ECS), Entropy corrections, Unstructured grids

© 2021 Aziz Madrane et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Discontinuous finite elements, Multi-dimensional slope limiter, Entropy Stable Scheme (ESS), Entropy Conservative Scheme (ECS), Entropy corrections, Unstructured grids