Admissible state and physical constraints preserving schemes for relativistic magnetohydrodynamic equations

TL;DR

This paper develops physical-constraints-preserving schemes for relativistic magnetohydrodynamics by analyzing the admissible state set, deriving explicit constraints, and establishing generalized Lax-Friedrichs splitting properties, ensuring solutions stay within physical bounds.

Contribution

It introduces new explicit forms of the admissible state set for RMHD, proves their convexity, and constructs PCP schemes that maintain physical constraints, including high-order schemes with divergence-free magnetic fields.

Findings

First-order PCP scheme is stable under CFL condition.

High-order PCP schemes are achievable with a limiter.

Numerical examples confirm theoretical results.

Abstract

This paper first studies the admissible state set $\mathcal G$ of relativistic magnetohydrodynamics (RMHD). It paves a way for developing physical-constraints-preserving (PCP) schemes for RMHD equations with the solutions in $\mathcal G$. To overcome the difficulties arising from the extremely strong nonlinearities and no explicit formulas of the primitive variables and flux vectors with respect to the conservative vector, two equivalent forms of $\mathcal G$ with explicit constraints on the conservative vector are skillfully discovered. The first is derived by analyzing roots of several polynomials and transferring successively them, and further used to prove the convexity of $\mathcal G$ with the aid of semi-positive definiteness of the second fundamental form of a hypersurface. While the second is derived based on the convexity and then used to show the orthogonal invariance of $\mathcal G$. The Lax-Friedrichs (LxF) splitting property does not hold generally for nonzero magnetic field, but by a constructive inequality and pivotal techniques, we discover the generalized LxF splitting properties, combining the convex combination of some LxF splitting terms with a discrete divergence-free condition of magnetic field. Based on above analyses, several one- and two-dimensional PCP schemes are then studied. In 1D case, a first-order accurate LxF type scheme is first proved to be PCP under the CFL condition, and then high-order PCP schemes are proposed via a PCP limiter. In 2D case, the discrete divergence-free condition and PCP property are analyzed for a first-order LxF scheme, and two sufficient conditions are derived for high-order accurate PCP schemes. Our analysis reveals in theory for the first time that the discrete divergence-free condition is closely connected with the PCP property. Several numerical examples demonstrate theoretical findings and performance of numerical schemes.