High-order upwind schemes for multidimensional magnetohydrodynamics

A general method for constructing high-order upwind schemes for multidimensional magnetohydrodynamics (MHD), having as a main built-in condition the divergence-free constraint del . B = 0 for the magnetic field vector B, is proposed. The suggested procedure is based on consistency arguments, by taking into account the specific operator structure of MHD equations with respect to the reference Euler equations of gasdynamics. This approach leads in a natural way to a staggered representation of the B field numerical data in which the divergence-free condition in the cell-averaged form, corresponding to second-order accurate numerical derivatives, is exactly fulfilled. To extend this property to higher order schemes, we then give general prescriptions to satisfy a (r + 1)th order accurate del . B = 0 relation for any numerical B held having a rth order interpolation accuracy. Consistency arguments lead also to a proper formulation of the upwind procedures needed to integrate the induction equations, assuring the exact conservation in time of the divergence-free condition and the related continuity properties for the B vector components. As an application, a third-order code to simulate multidimensional MHD flows of astrophysical interest is developed using essentially nonoscillatory-based reconstruction algorithms. Several test problems to illustrate and validate the proposed approach are finally presented.