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

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