A Free, Fast, Simple and Efficient TVD MHD Code

TL;DR

This paper introduces a new, efficient, and simple TVD MHD code that achieves high resolution, maintains divergence-free magnetic fields, and is suitable for scalable parallel computing, demonstrated through tests on waves and shocks.

Contribution

The paper presents a novel 2D/3D TVD MHD code with second order accuracy, divergence-free magnetic field enforcement, and a compact, open-source implementation.

Findings

High resolution per grid cell achieved

Magnetic divergence constraint preserved to machine precision

Code performs well on MHD wave and shock tests

Abstract

We describe a numerical method to solve the magnetohydrodynamic (MHD) equations. The fluid variables are updated along each direction using the flux conservative, 2nd order, total variation diminishing (TVD), upwind scheme of Jin and Xin. The magnetic field is updated separately in two-dimensional advection-constraint steps. The electromotive force (EMF) is computed in the advection step using the TVD scheme, and this same EMF is used immediately in the constraint step in order to preserve \grad.B=0 without the need to store intermediate fluxes. Operator splitting is used to extend the code to three dimensions, and Runge-Kutta is used to get second order accuracy in time. The advantages of this code are high resolution per grid cell, second order accuracy in space and time, enforcement of the \grad.B=0 constraint to machine precision, no memory overhead, speed, and simplicity. A 3-D Fortran implementation less than 400 lines long is made freely available. We also implemented a fully scalable message-passing parallel MPI version. We present tests of the code on MHD waves and shocks.