Numerical evolution of the resistive relativistic magnetohydrodynamic equations: a minimally implicit Runge-Kutta scheme

Abstract

We present the Minimally-Implicit Runge-Kutta (MIRK) methods for the numerical evolution of the resistive relativistic magnetohydrodynamic (RRMHD) equations, following the approach proposed by Komissarov (2007) of an augmented system of evolution equations to numerically deal with constraints. Previous approaches rely on Implicit-Explicit (IMEX) Runge-Kutta schemes; in general, compared to explicit schemes, IMEX methods need to apply the recovery (which can be very expensive computationally) of the primitive variables from the conserved ones in numerous additional times. Moreover, the use of an iterative process for the recovery could have potential convergence problems, increased by the additional number of required loops. In addition, the computational cost of the previous IMEX approach in comparison with the standard explicit methods is much higher. The MIRK methods are able to deal with stiff terms producing stable numerical evolutions, minimize the number of recoveries needed in comparison with IMEX methods, their computational cost is similar to the standard explicit methods and can actually be easily implemented in numerical codes which previously used explicit schemes. Two standard numerical tests are shown in the manuscript.