Total Variation Diminishing Scheme for Adiabatic and Isothermal Magnetohydrodynamics

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Abstract
In this paper a total variation diminishing (TVD) scheme is constructed for solving the equations of ideal adiabatic and isothermal MHD. It is based on an extremely efficient formulation of the MHD Riemann problems for either case. Piecewise linear interpolation is applied to the characteristic variables along with steepening of linearly degenerate characteristic fields. A predictor-corrector formulation is used to achieve second-order-accurate temporal update. An artificial viscosity and hyperviscosity are formulated using the characteristic variables. The viscosity and hyperviscosity are designed so that they never damage the TVD property. An accurate formulation of the divergence cleaning step is presented. This formulation is more accurate than the one that has been used so far. The scheme designed is second-order accurate in space and time. It has been implemented in the author's RIEMANN code for numerical MHD. A variety of test problems are presented. They test all aspects of numerical MHD including (1) handling of exotic wave structures that occur in MHD, (2) treatment of multiple discontinuities, (3) handling of very strong shocks, and (4) multidimensional problems. The scheme displays robust and accurate behavior in each case. An extremely efficient implementation has been achieved for massively parallel processor (MPP) machines displaying the ability of this scheme to sustain scalable, load-balanced performance in MPP environments.