# A high-order weighted finite difference scheme with a multi-state approximate Riemann solver for divergence-free magnetohydrodynamic simulations

## TL;DR

This paper introduces a high-order finite difference scheme for ideal magnetohydrodynamics that ensures shock-capturing, divergence-free magnetic fields, and high accuracy through advanced interpolation, Riemann solvers, and a novel constrained transport method.

## Contribution

It presents a new high-order finite difference scheme with a multi-state Riemann solver and a modified constrained transport method for divergence-free magnetic fields in MHD simulations.

## Key findings

- The scheme achieves high-order accuracy and shock-capturing capabilities.
- It maintains divergence-free magnetic fields with a new CT variant.
- Benchmark tests demonstrate improved solution quality.

## Abstract

We design a conservative finite difference scheme for ideal magnetohydrodynamic simulations that attains high-order accuracy, shock-capturing, and divergence-free condition of the magnetic field. The scheme interpolates pointwise physical variables from computational nodes to midpoints through a high-order nonlinear weighted average. The numerical flux is evaluated at the midpoint by a multi-state approximate Riemann solver for correct upwinding, and its spatial derivative is approximated by a high-order linear central difference to update the variables with designed order of accuracy and conservation. The magnetic and electric fields are defined at staggered grid points employed in the Constrained Transport (CT) method by Evans & Hawley (1988). We propose a new CT variant, in which the staggered electric field is evaluated so as to be consistent with the base one-dimensional Riemann solver and the staggered magnetic field is updated to be divergence-free as designed high-order finite difference representation. We demonstrate various benchmark tests to measure the performance of the present scheme. We discuss the effect of the choice of interpolation methods, Riemann solvers, and the treatment for the divergence-free condition on the quality of numerical solutions in detail.