Volume 12, Issue 1
High-Order Accurate Entropy Stable Finite Difference Schemes for One- and Two-Dimensional Special Relativistic Hydrodynamics

Junming Duan & Huazhong Tang

Adv. Appl. Math. Mech., 12 (2020), pp. 1-29.

Published online: 2019-12

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  • Abstract

This paper develops the high-order accurate entropy stable  finite difference schemes for one- and two-dimensional special relativistic hydrodynamic equations. The schemes are built on the entropy conservative flux and the weighted essentially non-oscillatory (WENO) technique as well as explicit Runge-Kutta time discretization. The key is to technically construct the {affordable} entropy conservative flux of the semi-discrete second-order accurate entropy conservative schemes satisfying the semi-discrete entropy equality for the found convex entropy pair. As soon as the entropy conservative flux is derived, the dissipation term can be added to give the semi-discrete entropy stable schemes satisfying the semi-discrete entropy inequality with the given convex entropy function. The WENO reconstruction for the scaled entropy variables and the high-order explicit Runge-Kutta time discretization are implemented to obtain the fully-discrete high-order entropy stable schemes.

  • Keywords

Entropy conservative scheme, entropy stable scheme, high order accuracy, finite difference scheme, special relativistic hydrodynamics.

  • AMS Subject Headings

65M10, 78A48

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

duanjm@pku.edu.cn (Junming Duan)

hztang@math.pku.edu.cn (Huazhong Tang)

  • BibTex
  • RIS
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@Article{AAMM-12-1, author = {Duan , Junming and Tang , Huazhong }, title = {High-Order Accurate Entropy Stable Finite Difference Schemes for One- and Two-Dimensional Special Relativistic Hydrodynamics}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2019}, volume = {12}, number = {1}, pages = {1--29}, abstract = {

This paper develops the high-order accurate entropy stable  finite difference schemes for one- and two-dimensional special relativistic hydrodynamic equations. The schemes are built on the entropy conservative flux and the weighted essentially non-oscillatory (WENO) technique as well as explicit Runge-Kutta time discretization. The key is to technically construct the {affordable} entropy conservative flux of the semi-discrete second-order accurate entropy conservative schemes satisfying the semi-discrete entropy equality for the found convex entropy pair. As soon as the entropy conservative flux is derived, the dissipation term can be added to give the semi-discrete entropy stable schemes satisfying the semi-discrete entropy inequality with the given convex entropy function. The WENO reconstruction for the scaled entropy variables and the high-order explicit Runge-Kutta time discretization are implemented to obtain the fully-discrete high-order entropy stable schemes.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0124}, url = {http://global-sci.org/intro/article_detail/aamm/13417.html} }
TY - JOUR T1 - High-Order Accurate Entropy Stable Finite Difference Schemes for One- and Two-Dimensional Special Relativistic Hydrodynamics AU - Duan , Junming AU - Tang , Huazhong JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 1 EP - 29 PY - 2019 DA - 2019/12 SN - 12 DO - http://dor.org/10.4208/aamm.OA-2019-0124 UR - https://global-sci.org/intro/article_detail/aamm/13417.html KW - Entropy conservative scheme, entropy stable scheme, high order accuracy, finite difference scheme, special relativistic hydrodynamics. AB -

This paper develops the high-order accurate entropy stable  finite difference schemes for one- and two-dimensional special relativistic hydrodynamic equations. The schemes are built on the entropy conservative flux and the weighted essentially non-oscillatory (WENO) technique as well as explicit Runge-Kutta time discretization. The key is to technically construct the {affordable} entropy conservative flux of the semi-discrete second-order accurate entropy conservative schemes satisfying the semi-discrete entropy equality for the found convex entropy pair. As soon as the entropy conservative flux is derived, the dissipation term can be added to give the semi-discrete entropy stable schemes satisfying the semi-discrete entropy inequality with the given convex entropy function. The WENO reconstruction for the scaled entropy variables and the high-order explicit Runge-Kutta time discretization are implemented to obtain the fully-discrete high-order entropy stable schemes.

Junming Duan & Huazhong Tang. (2019). High-Order Accurate Entropy Stable Finite Difference Schemes for One- and Two-Dimensional Special Relativistic Hydrodynamics. Advances in Applied Mathematics and Mechanics. 12 (1). 1-29. doi:10.4208/aamm.OA-2019-0124
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