Locally Divergence-preserving Upwind Finite Volume Schemes for Magnetohydrodynamic Equations

A main issue in nonstationary, compressible magnetohydrodynamic (MHD) simulations is controlling the divergence of the magnetic flux. This paper presents a general procedure showing how to modify the intercell fluxes in a conservative MHD finite volume code such that the scheme becomes locally divergence preserving. That is, a certain discrete divergence operator vanishes exactly during the entire simulation, which results in the suppression of any divergence error. The procedure applies to arbitrary finite volume schemes provided they are based on intercell fluxes. We deduce the necessary modifications for numerical methods based on rectangles and triangles and present numerical experiments with the new schemes. The theoretical justification of the schemes is given in two independent ways. One way starts with the discrete divergence operator that has to be preserved and modifies the fluxes accordingly. The second way uses a finite element reconstruction via Nedelec elements. Both methods lead to equivalent numerical methods.