High-order accurate entropy stable finite difference schemes for the shallow water magnetohydrodynamics

JOURNAL OF COMPUTATIONAL PHYSICS

卷:431

Article

Duan, Junming¹  Tang, Huazhong¹ 

[1] Peking Univ, Ctr Appl Phys & Technol, Sch Math Sci, HEDPS & LMAM, Beijing 100871, Peoples R China

关键词: Entropy conservative scheme;    Entropy stable scheme;    High-order accuracy;    Positivity preserving;    Finite difference scheme;    Shallow water magnetohydrodynamics;   

DOI  :  10.1016/j.jcp.2021.110136

来源: Elsevier

【 摘 要 】

This paper develops the high-order accurate entropy stable (ES) finite difference schemes for the shallow water magnetohydrodynamic (SWMHD) equations. They are built on the numerical approximation of the modified SWMHD equations with the Janhunen source term. First, the second-order accurate well-balanced semi-discrete entropy conservative (EC) schemes are constructed, satisfying the entropy identity for the given convex entropy function and preserving the steady states of the lake at rest (with zero magnetic field). The key is to match both discretizations for the fluxes and the non-flat river bed bottom and Janhunen source terms, and to find the affordable EC fluxes of the second order EC schemes. Next, by using the second-order EC schemes as building block, high order accurate well-balanced semi-discrete EC schemes are proposed. Then, the high order accurate well-balanced semi-discrete ES schemes are derived by adding a suitable dissipation term to the EC scheme with the WENO reconstruction of the scaled entropy variables in order to suppress the numerical oscillations of the EC schemes. After that, the semi-discrete schemes are integrated in time by using the high-order strong stability preserving explicit Runge-Kutta schemes to obtain the fully-discrete high-order wellbalanced schemes. The ES property of the Lax-Friedrichs flux is also proved and then the positivity-preserving ES schemes are studied by using the positivity-preserving flux limiters. Finally, extensive numerical tests are conducted to validate the accuracy, the wellbalanced, ES and positivity-preserving properties, and the ability to capture discontinuities of our schemes. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

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