# A Constrained Transport Method for the Solution of the Resistive Relativistic MHD Equations

## TL;DR

This paper introduces a new Godunov-type numerical method for resistive relativistic MHD that ensures divergence-free magnetic fields and charge conservation, improving stability and accuracy over existing methods.

## Contribution

The paper presents a novel constrained transport scheme with an efficient implicit-explicit Runge-Kutta method and a five-wave Riemann solver for resistive relativistic MHD.

## Key findings

- Method is more stable and robust than divergence cleaning approaches.
- Employs a less diffusive Riemann solver for better accuracy.
- Numerical benchmarks demonstrate superior performance.

## Abstract

We describe a novel Godunov-type numerical method for solving the equations of resistive relativistic magnetohydrodynamics. In the proposed approach, the spatial components of both magnetic and electric fields are located at zone interfaces and are evolved using the constrained transport formalism. Direct application of Stokes' theorem to Faraday's and Ampere's laws ensures that the resulting discretization is divergence-free for the magnetic field and charge-conserving for the electric field. Hydrodynamic variables retain, instead, the usual zone-centred representation commonly adopted in finite-volume schemes. Temporal discretization is based on Runge-Kutta implicit-explicit (IMEX) schemes in order to resolve the temporal scale disparity introduced by the stiff source term in Ampere's law. The implicit step is accomplished by means of an improved and more efficient Newton-Broyden multidimensional root-finding algorithm. The explicit step relies on a multidimensional Riemann solver to compute the line-averaged electric and magnetic fields at zone edges and it employs a one-dimensional Riemann solver at zone interfaces to update zone-centred hydrodynamic quantities. For the latter, we introduce a five-wave solver based on the frozen limit of the relaxation system whereby the solution to the Riemann problem can be decomposed into an outer Maxwell solver and an inner hydrodynamic solver. A number of numerical benchmarks demonstrate that our method is superior in stability and robustness to the more popular charge-conserving divergence cleaning approach where both primary electric and magnetic fields are zone-centered. In addition, the employment of a less diffusive Riemann solver noticeably improves the accuracy of the computations.