On 2D gas dynamics simulation using RKDG method on structured rectangular meshes

Abstract

In this paper, we describe the variant of the Runge - Kutta Discontinuous Galerkin (RKDG) method for numerical simulation of 2D gas dynamics flows on structured rectangular meshes. The RKDG algorithm was implemented with C++ code based on the experience in 1D case implementation and verified on simple tests. The advantage of the RKDG method over the most popular finite volume method (FVM) is discussed: three basis functions being applied in the framework of the RKDG approach lead to a considerable decrease of the numerical dissipation rate with respect to FVM. The numerical code contains implementations of Lax - Friedrichs, HLL and HLLC numerical fluxes, KXRCF troubled cell indicator and WENO_S and MUSCL slope limiters. Results of the acoustic pulse simulation on a sufficiently coarse mesh with the piecewise-linear approximation show a good agreement with the analytical solution in contrast to the FVM numerical solution. For the Sod problem results of the shock wave, rarefaction wave and the contact discontinuity propagation illustrate the dependence of the scheme resolution on the numerical fluxes, troubled cell indicator and limitation technique choice. The numerical scheme with MUSCL slope limiter has the higher numerical dissipation in comparison to one with WENO_S limiter. It is shown, that in some cases KXRCF troubled cell indicator doesn’t detect numerical solution non-monotonicity.

Keywords

• rkdg-метод
• уравнения эйлера
• muscl-схема
• weno-схема
• лимитер
• индикатор