Relativistic Magnetohydrodynamics: Renormalized eigenvectors and full wave decomposition Riemann solver

TL;DR

This paper develops a complete set of renormalized eigenvectors for relativistic MHD equations, enabling a robust full wave decomposition Riemann solver that improves accuracy over simpler methods, especially in multidimensional simulations.

Contribution

It introduces a novel renormalization of eigenvectors for relativistic MHD, allowing a full wave decomposition Riemann solver applicable to degenerate states, enhancing numerical robustness and accuracy.

Findings

The solver is highly robust in standard tests.

It is less diffusive than HLL and HLLC solvers.

Efficiency improves in multidimensional simulations.

Abstract

We obtain renormalized sets of right and left eigenvectors of the flux vector Jacobians of the relativistic MHD equations, which are regular and span a complete basis in any physical state including degenerate ones. The renormalization procedure relies on the characterization of the degeneracy types in terms of the normal and tangential components of the magnetic field to the wavefront in the fluid rest frame. Proper expressions of the renormalized eigenvectors in conserved variables are obtained through the corresponding matrix transformations. Our work completes previous analysis that present different sets of right eigenvectors for non-degenerate and degenerate states, and can be seen as a relativistic generalization of earlier work performed in classical MHD. Based on the full wave decomposition (FWD) provided by the the renormalized set of eigenvectors in conserved variables, we have also developed a linearized (Roe-type) Riemann solver. Extensive testing against one- and two-dimensional standard numerical problems allows us to conclude that our solver is very robust. When compared with a family of simpler solvers that avoid the knowledge of the full characteristic structure of the equations in the computation of the numerical fluxes, our solver turns out to be less diffusive than HLL and HLLC, and comparable in accuracy to the HLLD solver. The amount of operations needed by the FWD solver makes it less efficient computationally than those of the HLL family in one-dimensional problems. However its relative efficiency increases in multidimensional simulations.