Classical and semirelativistic magnetohydrodynamics with anisotropic ion pressure

We study the magnetohydrodynamics (MHD) equations with anisotropic ion pressure and isotropic electron pressure under both the classical and semirelativistic approximations in order to develop a numerical model. The dispersion relation as well as the characteristic wave speeds are derived. In addition to the exact wave speed solutions, we also provide efficient approximate formulas for the semirelativistic magnetosonic speeds. The equations are discretized with the Rusanov and Harten-Lax-van Leer numerical schemes and implemented into the BATS-R-US MHD code. We perform a set of verification tests.