An Unsplit, Cell-Centered Godunov Method for Ideal MHD

TL;DR

This paper introduces a second-order, unsplit Godunov method for multidimensional ideal MHD that maintains divergence-free magnetic fields and demonstrates high accuracy and stability across various tests.

Contribution

The paper presents a novel unsplit, cell-centered Godunov algorithm for ideal MHD that effectively enforces divergence-free magnetic fields using a discrete projection and filtering techniques.

Findings

Second-order accuracy for smooth solutions

Converges correctly for shock problems

Stable and accurate with non-solenoidal fields

Abstract

We present a second-order Godunov algorithm for multidimensional, ideal MHD. Our algorithm is based on the unsplit formulation of Colella (J. Comput. Phys. vol. 87, 1990), with all of the primary dependent variables centered at the same location. To properly represent the divergence-free condition of the magnetic fields, we apply a discrete projection to the intermediate values of the field at cell faces, and apply a filter to the primary dependent variables at the end of each time step. We test the method against a suite of linear and nonlinear tests to ascertain accuracy and stability of the scheme under a variety of conditions. The test suite includes rotated planar linear waves, MHD shock tube problems, low-beta flux tubes, and a magnetized rotor problem. For all of these cases, we observe that the algorithm is second-order accurate for smooth solutions, converges to the correct weak solution for problems involving shocks, and exhibits no evidence of instability or loss of accuracy due to the possible presence of non-solenoidal fields.