Divergence-free-preserving high-order schemes for magnetohydrodynamics: An artificial magnetic resistivity method

This paper proposes a new strategy that is very simple, divergence-free, high-order accurate, yet has an effective discontinuous-capturing capability for simulating compressible magnetohydrodynamics (MHD). The new strategy is to construct artificial diffusion terms in a physically-consistent manner, such that the artificial 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 physically-consistent artificial diffusion terms are built into the induction equations in a conservation law form at a partial-differential-equation level. The proposed method may be viewed as adding an artificial magnetic resistivity to the induction equations, and is inherently divergence-free both ideal and resistive MHD, with and without shock waves, and also both inviscid and viscous flows. The method is based on finite difference method with co-located variable arrangement, and we show that any linear finite difference scheme in an arbitrary order of accuracy can be used to discretize the modified governing equations to ensures the divergence-free and the global conservation constraints numerically at the discretization level. The artificial magnetic resistivity is designed to localize automatically in regions of discontinuity in the curl of magnetic field and vanish wherever the flow is sufficiently smooth with respect to the grid scale, thereby maintaining the desirable high-order accuracy of the employed discretization scheme in smooth regions. Two-dimensional smooth and non-smooth ideal MHD problems are considered to validate the capability of the proposed method.