A Fourth-Order Finite Volume Scheme for Resistive Relativistic Magnetohydrodynamics

TL;DR

This paper introduces a high-order finite-volume numerical scheme for resistive relativistic magnetohydrodynamics, combining advanced spatial reconstruction, Riemann solvers, and implicit-explicit time stepping to improve accuracy and stability.

Contribution

It develops a genuinely 4th-order accurate method for resistive RMHD, integrating WENO-Z reconstruction, MHLLC flux, and IMEX Runge-Kutta schemes for the first time.

Findings

Achieves 4th-order spatial and temporal accuracy in tests.

Demonstrates robustness and stability in numerical benchmarks.

Effectively handles stiffness from source terms.

Abstract

We present a finite-volume, genuinely 4th-order accurate numerical method for solving the equations of resistive relativistic magnetohydrodynamics (Res-RMHD) in Cartesian coordinates. In our formulation, the magnetic field is evolved in time in terms of face-average values via the constrained-transport method while the remaining variables (density, momentum, energy and electric fields) are advanced as cell volume-averages. Spatial accuracy employs 5th-order accurate WENO-Z reconstruction from point values (as described in a companion paper) to obtain left and right states at zone interfaces. Explicit flux evaluation is carried out by solving a Riemann problem at cell interfaces, using the Maxwell-Harten-Lax-van Leer with contact wave resolution (MHLLC). Time stepping is based on the implicit-explicit (IMEX) Runge-Kutta (RK) methods, of which we consider both the 3rd-order strong stability preserving SSP3(4,3,3) and a recent 4th-order additive RK scheme, to cope with the stiffness introduced by the source term in Ampere's law. Numerical benchmarks are presented in order to assess the accuracy and robustness of our implementation.