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

TL;DR

This paper introduces the MIRK methods for efficiently evolving resistive relativistic magnetohydrodynamic equations, reducing computational costs and recovery steps compared to traditional IMEX schemes, and demonstrating their stability and ease of implementation.

Contribution

The paper proposes Minimally-Implicit Runge-Kutta (MIRK) methods that improve upon IMEX schemes by reducing computational cost and recovery steps for resistive relativistic MHD simulations.

Findings

MIRK methods handle stiff terms with stability.

Reduced number of primitive variable recoveries.

Comparable computational cost to explicit schemes.

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.