Shock Capturing with Higher-Order, PDE-Based Artificial Viscosity

Abstract

Artificial viscosity can be combined with a higher-order discontinuous Galerkin (DG) discretization to resolve a shock layer within a single cell. However, when a piecewiseconstant artificial viscosity model is employed with an otherwise higher-order approximation, element-to-element variations in the artificial viscosity arising at the shock induce oscillations in state gradients and pollute the downstream flow. To alleviate these difficulties, this work proposes a new higher-order, state-based artificial viscosity with an associated governing PDE. In the governing PDE, the shock sensor acts as a forcing term, driving the artificial viscosity to a non-zero value where it is necessary. The decay rate of the higher-order solution modes and edge-based jumps are both shown to be reliable shock indicators. This new approach leads to a smooth, higher-order representation of the artificial viscosity, that evolves in time with the solution. Additionally, an artificial dissipation operator that preserves total enthalpy is introduced. The combination of higher-order, PDE-based artifical viscosity and enthalpy-preserving dissipation operator is shown to overcome the disadvantages of the piecewise-constant artificial viscosity, while achieving greater robustness on flows with strong shocks.