Uniformly High Order Discontinuous Galerkin Gas Kinetic Scheme for Compressible flows

TL;DR

This paper develops a one-stage uniformly high-order discontinuous Galerkin gas kinetic scheme (DG-HGKS) for compressible Euler flows by combining a fully discrete DG formulation with time-expansion of numerical fluxes and ADER-like spatial derivative replacement, enabling arbitrarily high spatial-temporal accuracy. It integrates a KXRCF-based detector with SHWENO limiting to suppress non-physical oscillations in discontinuous regions while maintaining compact stencils and fewer troubled cells than RKDG approaches. The method uses a BGK-based flux formulation that naturally dissipates near shocks and employs Gaussian quadrature and Lax-Wendroff/ADER techniques to achieve high-order time integration within a one-stage framework. Numerical tests in 1D and 2D demonstrate robust accuracy, reduced limiter activity, and favorable comparisons to CEHGKS and RKDG, highlighting improved efficiency and potential extensions to viscous flows.

Abstract

In this paper, a uniformly high-order discontinuous Galerkin gas kinetic scheme (DG-HGKS) is proposed to solve the Euler equations of compressible flows. The new scheme is an extension of the one-stage compact and efficient high-order GKS (CEHGKS, Li et al. , 2021. J. Comput. Phys. 447, 110661) in the finite volume framework. The main ideas of the new scheme consist of two parts. Firstly, starting from a fully discrete DG formulation, the numerical fluxes and volume integrals are expanded in time. Secondly, the time derivatives are replaced by spatial derivatives using the techniques in CEHGKS. To suppress the non-physical oscillations in the discontinuous regions while minimizing the number of "troubled cells", an effective limiter strategy compatible with the new scheme has been developed by combining the KXRCF indicator and the SHWENO reconstruction technique. The new scheme can achieve arbitrary high-order accuracy in both space and time, thereby breaking the previous limitation of no more than third-order accuracy in existing one-stage DG-HGKS schemes. Numerical tests in 1D and 2D have demonstrated the robustness and effectiveness of the scheme.

Figures (30)

• Figure 4.1: The density distribution of Example 4.2 \ref{['sod']} with $N=200.$ Left top: second-order($p^1$), right top: third-order($p^2$), left bottom: fourth-order($p^3$), right bottom: fifth-order($p^4$).
• Figure 4.2: The time history of the troubled cells for Example 4.2 \ref{['sod']} with $N=200.$ DG-HGKS with SHWENO limiter. From left to right: second-order($p^1$), third-order($p^2$), fourth-order($p^3$), fifth-order($p^4$).
• Figure 4.3: The density distribution of Example 4.2 \ref{['lax']} with $N=200.$ Left top: second-order($p^1$), right top: third-order($p^2$), left bottom: fourth-order($p^3$), right bottom: fifth-order($p^4$).
• Figure 4.4: The time history of the troubled cells for Example 4.2 \ref{['lax']} with $N=200.$ DG-HGKS with SHWENO limiter. From left to right: second-order($p^1$), third-order($p^2$), fourth-order($p^3$), fifth-order($p^4$).
• Figure 4.5: The density distribution of Example 4.3 \ref{['so']} with $N=200.$ Left top: second-order($p^1$), right top: third-order($p^2$), left bottom: fourth-order($p^3$), right bottom: fifth-order($p^4$).