An Entropy Stable Finite Volume Scheme for the Equations of Shallow Water Magnetohydrodynamics

TL;DR

This paper introduces an entropy stable finite volume scheme for shallow water magnetohydrodynamics equations, ensuring entropy preservation and robustness in shock-prone scenarios through a novel flux function and dissipation treatment.

Contribution

It presents a new entropy stable finite volume method with an analytical flux function that exactly preserves entropy for SWMHD equations, including a special source term treatment.

Findings

The scheme conserves entropy exactly in numerical tests.

The method demonstrates robustness in shock development scenarios.

Numerical results confirm theoretical entropy stability.

Abstract

In this work, we design an entropy stable, finite volume approximation for the shallow water magnetohydrodynamics (SWMHD) equations. The method is novel as we design an affordable analytical expression of the numerical interface flux function that exactly preserves the entropy, which is also the total energy for the SWMHD equations. To guarantee the discrete conservation of entropy requires a special treatment of a consistent source term for the SWMHD equations. With the goal of solving problems that may develop shocks, we determine a dissipation term to guarantee entropy stability for the numerical scheme. Numerical tests are performed to demonstrate the theoretical findings of entropy conservation and robustness.