Characteristic Decomposition for Relativistic Numerical Simulations: II. Magnetohydrodynamics

TL;DR

This work delivers a complete characteristic decomposition for general-relativistic magnetohydrodynamics by introducing a quasi-invertible transformation that links comoving variables $(u^a,b^a)$ to Eulerian variables $(v^a,B^a)$ and extends to conserved-variable representations. It derives explicit right and left eigenvectors for entropy, Alfvén, and magnetosonic modes in both nonconservative and divergence-cleaning formulations, including their Eulerian and conserved-variable projections. A key outcome is that the GRMHD eigensystem is simpler than previously thought and amenable to high-fidelity numerical methods, such as full-wave Riemann solvers, with straightforward implementation in codes like SpECTRE. The work also analyzes the divergence-cleaning variant, showing the retained similarity of eigenstructure to Anile’s formulation while introducing scalar modes that restore strong hyperbolicity. These results enable robust, hyperbolic GRMHD simulations and pave the way for improved boundary conditions and flux calculations in relativistic astrophysical applications.

Abstract

The characteristic decomposition for GRMHD in the comoving frame of the fluid has been known for a long time. However, it has not been known in the coordinate frame of the simulation and in terms of the conserved variables evolved in typical numerical simulations. This paper applies the method of quasi-invertible transformations developed in Paper I to derive this decomposition. Among other benefits, this will allow us to use the most accurate known computational methods, such as full-wave Riemann solvers. The results turn out to be simpler than expected based on earlier attempts.