Robust flux limiting of the Runge-Kutta Discontinuous Galerkin method for advection-dominated multiphysics plasma simulations

The WARPXM code (Shumlak et al., CPC 2011) implements the Runge-Kutta Discontinuous Galerkin (RKDG) method to solve a variety of hyperbolic-parabolic PDEs. These include the resistive MHD equations and the 5N-moment fluid equations with Braginskii transport. Source terms such as ionization and collisional terms are also included. The RKDG method has several advantages for such multiphysics problems, including the ability to achieve exact conservation and arbitrarily high-order accuracy. However, when using the RKDG method with a high-order spatial discretization, oscillations can appear at discontinuities in advection-dominated regions of the solution. Furthermore, maintaining positivity and avoiding the appearance of large, non-physical wave speeds are of key importance for applications, but are complicated by these oscillations. To address these shortcomings, we develop a flux-limiting procedure for RKDG methods based on the parametrized positivity-preserving flux limiter of Xiong et al. (JSC 2016). By computing speculative timesteps of the cell average based on a first-order DG method with a monotonic flux, we limit the solution to one which preserves positivity of density and pressure. High-order accuracy in time is achieved through an SSPRK method. We implement the limiter in the WARPXM code, and demonstrate its effectiveness on both classical test problems and stronger shocks. Special attention is paid to the treatment of diffusive terms and source terms in the method.