A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations

In this paper, we construct a high-order discontinuous Galerkin (DG) method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics (VRMHD). To control the divergence error in the magnetic field, both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD. Rigorous theoretical analyses are presented for one-dimensional and multi-dimensional DG schemes, respectively, showing that the scheme can maintain the positivity-preserving (PP) property under some CFL conditions when combined with the strong-stability-preserving time discretization. Then, general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes. Numerical tests demonstrate the effectiveness of the proposed schemes.