A study of an artificial viscosity technique for high-order discontinuous Galerkin methods

Abstract

Prediction of heat loads during hypersonic re-entry is of great interest in space exploration and in the topic of space debris as well. To date, there is no accurate method to reproduce either experimentally or numerically the physics in re-entry conditions. On the numerical side, high-order discontinuous Galerkin methods have potential to improve results achieved with state-of-the-art finite volume methods. However, they suffer from Gibbs-type oscillations around discontinuities such as shocks. This work uses artificial viscosity to tackle this issue.
The artificial viscosity method used in this work contains three user-defined parameters which specify the magnitude, location and width of the artificial viscosity. A parametric study of these parameters on a steady inviscid wedge test case at Mach 2 showed that tuning their values is a non-trivial task and it appeared that in some cases shock smoothness and shock thickness need to be traded-off. Furthermore, it was observed that sufficient refinement in mesh size or polynomial order was needed to obtain satisfying results. The Sod test case was considered to test the method in an unsteady inviscid case. Good results were obtained, although the shock was better resolved than the contact discontinuity. The method was also tested on another unsteady inviscid test case, the Shu-Osher problem. The latter is more challenging due to the shock strength varying in time. Specifying the artificial viscosity parameters was challenging since the set had to account for variation of the shock strength in time. The test case also highlighted the need for a method that keeps variables within their physical bounds.
Prediction of the heat flux on the surface of a half-cylinder in a hypersonic flow was attempted. However, this was unsuccessful and the Mach number was eventually reduced to 3.25. Strong non-physical oscillations appeared in the heat flux profile even with smooth contour plots and pressure profile at the surface of the half-cylinder. The key to obtain a smooth heat flux profile was to ensure no artificial viscosity at all is inserted in the boundary layer. Also, a symmetric mesh was needed to obtain a symmetric profile. Overall, the heat flux results showed that it is possible to simulate this challenging test case with discontinuous Galerkin and a combination of simple methods, including the Lax-Friedrichs flux and a basic smoothing of the artificial viscosity field.
A new strategy to tune the user-defined parameter specifying the location of the artificial viscosity was implemented and tested on the Shu-Osher and half-cylinder test cases. It was found to be greatly beneficial in terms of stability and user-friendliness, but did not fully eliminate the issues in simulating these challenging test cases. This strategy allowed to run the half-cylinder test case beyond the Lax-Friedrichs flux and a polynomial order of 1. The heat flux profile with SLAU came very close to that with Lax-Friedrichs, whereas the difference was significant between first and second orders.