High-order accurate entropy stable nodal discontinuous Galerkin schemes for the ideal special relativistic magnetohydrodynamics

TL;DR

This paper develops high-order entropy stable nodal discontinuous Galerkin schemes for ideal special relativistic magnetohydrodynamics, ensuring accuracy, stability, and improved magnetic field handling through novel flux design.

Contribution

It introduces a new entropy conservative flux for RMHD that maintains the zero parallel magnetic component and constructs high-order entropy stable DG schemes based on this flux.

Findings

Schemes achieve high-order accuracy and stability.

Numerical tests validate the schemes' ability to capture discontinuities.

The new flux reduces error in the parallel magnetic component in 1D tests.

Abstract

This paper studies high-order accurate entropy stable nodal discontinuous Galerkin (DG) schemes for the ideal special relativistic magnetohydrodynamics (RMHD). It is built on the modified RMHD equations with a particular source term, which is analogous to the Powell's eight-wave formulation and can be symmetrized so that an entropy pair is obtained. We design an affordable fully consistent two-point entropy conservative flux, which is not only consistent with the physical flux, but also maintains the zero parallel magnetic component, and then construct high-order accurate semi-discrete entropy stable DG schemes based on the quadrature rules and the entropy conservative and stable fluxes. They satisfy the semidiscrete entropy inequality for the given entropy pair and are integrated in time by using the high-order explicit strong stability preserving Runge-Kutta schemes to get further the fully-discrete nodal DG schemes. Extensive numerical tests are conducted to validate the accuracy and the ability to capture discontinuities of our schemes. Moreover, our entropy conservative flux is compared to an existing flux through some numerical tests. The results show that the zero parallel magnetic component in the numerical flux can help to decrease the error in the parallel magnetic component in one-dimensional tests, but two entropy conservative fluxes give similar results since the error in the magnetic field divergence seems dominated in the two-dimensional tests.