Approximate Riemann Solvers for the Cosmic Ray Magnetohydrodynamical Equations

TL;DR

This paper develops an approximate Riemann solver for cosmic-ray magnetohydrodynamics, improving shock resolution and numerical accuracy by reformulating the equations in a fully conservative form and validating it against known solutions.

Contribution

It introduces a new conservative formulation of CR MHD equations and derives an approximate Riemann solver that accurately captures shocks and improves numerical simulations.

Findings

The new solver reproduces the Riemann solution for CR shocks.

Solutions in the conservative form converge to the Riemann solution with second-order accuracy.

The method sharply resolves fast and slow shocks in magnetized media.

Abstract

We analyze the cosmic-ray magnetohydrodynamic (CR MHD) equations to improve the numerical simulations. We propose to solve them in the fully conservation form, which is equivalent to the conventional CR MHD equations. In the fully conservation form, the CR energy equation is replaced with the CR "number" conservation, where the CR number density is defined as the three fourths power of the CR energy density. The former contains an extra source term, while latter does not. An approximate Riemann solver is derived from the CR MHD equations in the fully conservation form. Based on the analysis, we propose a numerical scheme of which solutions satisfy the Rankine-Hugoniot relation at any shock. We demonstrate that it reproduces the Riemann solution derived by Pfrommer et al. (2006) for a 1D CR hydrodynamic shock tube problem. We compare the solution with those obtained by solving the CR energy equation. The latter solutions deviate from the Riemann solution seriously, when the CR pressure dominates over the gas pressure in the post-shocked gas. The former solutions converge to the Riemann solution and are of the second order accuracy in space and time. Our numerical examples include an expansion of high pressure sphere in an magnetized medium. Fast and slow shocks are sharply resolved in the example. We also discuss possible extension of the CR MHD equations to evaluate the average CR energy.