High-order divergence-free method for compressible MHD with shock waves

This paper presents a new strategy that is very simple, divergence-free-preserving, high- resolution, and high-order-accurate for simulation of compressible magnetohydrodynamics flows with shock waves. The proposed method is to explicitly adds physically-consistent artificial diffusion terms to the induction equations in a conservation law form in order to robustly capture numerical discontinuities in the magnetic field. We analytically show that the physically-consistent artificial diffusion terms act as a diffusion term only in the curl of magnetic field to capture numerical discontinuities in the magnetic field while not affecting the divergence field (thus maintaining divergence-free constraint). The artificial terms can be easily constructed (and also easily implemented in an existing code) by augmenting the physical magnetic resistivity by the artificial magnetic resistivity. Any linear differencing scheme in an arbitrary order of accuracy can be used to discretize the modified governing equations associated with the artificial magnetic resistivity to satisfy the divergence-free constraints numerically at the discretization level (sixth-order compact differencing scheme is used in this study). The artificial magnetic resistivity is dynamically localized only near the discontinuities and automatically vanishes in smooth regions, thus preserving the non- dissipative and high-wavenumber-resolution characteristics of high-order accurate compact scheme in smooth regions.