Localized Artificial Viscosity Stabilization of Discontinuous Galerkin Methods for Nonhydrostatic Mesoscale Atmospheric Modeling

Abstract

Gibbs oscillation can show up near flow regions with strong temperature gradients in the numerical simulation of nonhydrostatic mesoscale atmospheric flows when using the high-order discontinuous Galerkin (DG) method. The authors propose to incorporate flow-feature-based localized Laplacian artificial viscosity in the DG framework to suppress the spurious oscillation in the vicinity of sharp thermal fronts but not to contaminate the smooth flow features elsewhere. The parameters in the localized Laplacian artificial viscosity are modeled based on both physical criteria and numerical features of the DG discretization. The resulting numerical formulation is first validated on several shock-involved test cases, including a shock discontinuity problem with the one-dimensional Burger’s equation, shock–entropy wave interaction, and shock–vortex interaction. Then the efficacy of the developed numerical formulation on stabilizing thermal fronts in nonhydrostatic mesoscale atmospheric modeling is demonstrated by two benchmark test cases: the rising thermal bubble problem and the density current problem. The results indicate that the proposed flow-feature-based localized Laplacian artificial viscosity method can sharply detect the nonsmooth flow features, and stabilize the DG discretization nearby. Furthermore, the numerical stabilization method works robustly for a wide range of grid sizes and polynomial orders without parameter tuning in the localized Laplacian artificial viscosity.