Relativistic Magnetohydrodynamics: Renormalized Eigenvectors and Full Wave Decomposition Riemann Solver

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 twodimensional 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.

Building a numerical relativistic non-ideal magnetohydrodynamics code for astrophysical applications

“…Our results (Fig. 1 right) are comparable with the those obtained with our relativistic ideal MHD code (Leismann et al 2005, Anton et al 2010.…”

Grid-based Methods in Relativistic Hydrodynamics and Magnetohydrodynamics

“…Moreover, the MHD equations exhibit degeneracies in the sense that two or more eigenvalues may coincide, i.e., the set of equations is not strictly hyperbolic. The degeneracy conditions in RMHD have been analyzed by Komissarov (1999a), and more recently by Antón et al (2010). They coincide with those for Newtonian flows in the fluid rest frame.…”

“…Only if the tangential component of the fluid velocity vanishes, one recovers the classical behavior. The degeneracies can be visualized with the help of the characteristic wave speed diagrams (Antón et al, 2010), equivalent to the phase speed diagrams for linear perturbations . These diagrams show the normal speed of planar wave fronts propagating in different directions, the speed given by the distance between the origin and the normal speed surface along the corresponding direction.…”

Magnetohydrodynamics in stationary and axisymmetric spacetimes: A fully covariant approach

“…General relativistic magnetohydrodynamics (GRMHD) is a rapidly developing field of modern astrophysics [1][2][3], driven by numerous observations of accretion disks around black holes [4], jets in active galactic nuclei or microquasars [5,6], gamma ray bursts, hypernovae, pulsars [1] and strongly magnetized neutrons stars (magnetars). In a first approximation, all these systems are stationary and axisymmetric.…”