A simple unsplit Godunov method for multidimensional MHD

We describe a numerical algorithm based on Godunov methods for integrating the equations of compressible magnetohydrodynamics (MHD) in multidimensions. It combines a simple, dimensionally-unsplit integration method with the constrained transport (CT) discretization of the induction equation to enforce the divergence-free constraint. We present the results of a series of fully three-dimensional tests which indicate the method is second-order accurate for smooth solutions in all MHD wave families, and captures shocks, contact and rotational discontinuities well. However, it is also more diffusive than other more complex unsplit integrators combined with CT. Thus, the primary advantage of the method is its simplicity. It does not require a characteristic tracing step to construct interface values for the Riemann solver, it is straightforward to extend with additional physics, and it is suitable for use with nested and adaptive meshes. The method is implemented as one of two dimensionally unsplit MHD integrators in the Athena code, which is freely available for download from the web. (C) 2008 Elsevier B.V. All rights reserved.