Affordable, Entropy Conserving and Entropy Stable Flux Functions for the Ideal MHD Equations

TL;DR

This paper introduces an affordable, entropy stable finite volume method for ideal MHD equations that preserves entropy discretely and ensures robustness in shock scenarios.

Contribution

It presents a novel analytical flux function that conserves entropy and a dissipation mechanism for entropy stability at shocks.

Findings

Numerical tests confirm entropy conservation.

The scheme is robust for shock problems.

The method is computationally affordable.

Abstract

In this work, we design an entropy stable, finite volume approximation for the ideal magnetohydrodynamics (MHD) equations. The method is novel as we design an affordable analytical expression of the numerical interface flux function that discretely preserves the entropy of the system. To guarantee the discrete conservation of entropy requires the addition of a particular source term to the ideal MHD system. Exact entropy conserving schemes cannot dissipate energy at shocks, thus to compute accurate solutions to 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.