We present and compare third- as well as fifth-order accurate finite
difference schemes for the numerical solution of the compressible ideal MHD
equations in multiple spatial dimensions. The selected methods lean on four
different reconstruction techniques based on recently improved versions of the
weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving
(MP) schemes as well as slope-limited polynomial reconstruction. The proposed
numerical methods are highly accurate in smooth regions of the flow, avoid loss
of accuracy in proximity of smooth extrema and provide sharp non-oscillatory
transitions at discontinuities. We suggest a numerical formulation based on a
cell-centered approach where all of the primary flow variables are discretized
at the zone center. The divergence-free condition is enforced by augmenting the
MHD equations with a generalized Lagrange multiplier yielding a mixed
hyperbolic/parabolic correction, as in Dedner et al. (J. Comput. Phys. 175
(2002) 645-673). The resulting family of schemes is robust, cost-effective and
straightforward to implement. Compared to previous existing approaches, it
completely avoids the CPU intensive workload associated with an elliptic
divergence cleaning step and the additional complexities required by staggered
mesh algorithms. Extensive numerical testing demonstrate the robustness and
reliability of the proposed framework for computations involving both smooth
and discontinuous features.Comment: 32 pages, 14 figure, submitted to Journal of Computational Physics
(Aug 7 2009
