Linearized Formulation of the Riemann Problem for Adiabatic and Isothermal Magnetohydrodynamics

Abstract

In this paper we present a linearized formulation of the Riemann problem for ideal adiabatic and isothermal magnetohydrodynamic (MHD). Roe's property U is obtained. This ensures good capturing of shocks of arbitrary strength. A parameter vector is found which gives an almost consistent mean state, where consistency is defined in the sense of Roe. The eigenvalues and eigenvectors for the linearized Riemann problem for MHD have been explicitly constructed further facilitating numerical implementation. The method has been tested in the author's RIEMANN code for numerical MHD. In a companion paper on TVD methods for MHD, where this Riemann solver was used, we have shown that excellent results can be obtained. The consistency for this formulation is illustrated by the fact that the eigenvalues have the same symmetrical distribution around the Roe-averaged x-component of the velocity that the eigenvalues for ideal MHD have around the physical x-component of the velocity. The accuracy, simplicity, and noniterative nature of the linearized formulation of the MHD Riemann problem presented here make it especially suitable for use on massively parallel processor machines.