A Consistent Method of Characteristics for Multidimensional Magnetohydrodynamics

This paper introduces a previously unaddressed problem with performing multidimensional numerical MHD characterized by "explosive" growth (within a single time step) of weak magnetic fields to possibly dynamic strengths in the vicinity of strong velocity shear. This problem has been observed to occur in numerical algorithms which rely on the solution of the characteristic equations at the mesh interfaces. The solution to this problem is the subject of this paper and has led to an algorithm fundamentally different from those discussed previously. Specifically, at every point on the mesh, the magnetic induction terms are determined from the characteristic velocities which are evaluated by solving implicitly (in space) the characteristic equations for incompressible MHD (although the algorithm is perfectly suitable for compressible MHD). The implicit treatment of the characteristic equations sets this algorithm apart from previous efforts and is shown to be a necessary condition for multidimensional accuracy. It amounts to replacing the usual directional-split treatment of the induction and transport operators with a planar-split scheme.