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Comparison of the high-order Runge-Kutta discontinuous Galerkin method and gas-kinetic scheme for inviscid compressible flow simulations

Yixiao Wang, Xing Ji, Gang Chen, Kun Xu

TL;DR

This study benchmarks the high-order RKDG method and the Gas-Kinetic Scheme (GKS) for inviscid compressible flows, highlighting the trade-offs between RKDG’s high accuracy and sensitivity to limiters and GKS’s relaxed CFL constraints and physically grounded fluxes. By comparing across 1D and 2D Euler tests with both non-compact and compact reconstructions, the work shows that CGKS can achieve competitive or superior accuracy with fewer degrees of freedom in smooth regions and can handle shocks robustly, while RKDG maintains superior performance in certain discontinuous regimes. The results motivate a hybrid $h$-$p$ approach that blends DG and CGKS for unified treatment of smooth and sharp features, with future extensions to viscous flows and limiter robustness. Overall, the paper provides a detailed, fair comparison and practical insights into leveraging the strengths of both high-order methods for compressible gas dynamics.

Abstract

The Runge--Kutta discontinuous Galerkin (RKDG) method is a high-order technique for addressing hyperbolic conservation laws, which has been refined over recent decades and is effective in handling shock discontinuities. Despite its advancements, the RKDG method faces challenges, such as stringent constraints on the explicit time-step size and reduced robustness when dealing with strong discontinuities. On the other hand, the Gas-Kinetic Scheme (GKS) based on a high-order gas evolution model also delivers significant accuracy and stability in solving hyperbolic conservation laws through refined spatial and temporal discretizations. Unlike RKDG, GKS allows for more flexible CFL number constraints and features an advanced flow evolution mechanism at cell interfaces. Additionally, GKS' compact spatial reconstruction enhances the accuracy of the method and its ability to capture stable strong discontinuities effectively. In this study, we conduct a thorough examination of the RKDG method using various numerical fluxes and the GKS method employing both compact and non-compact spatial reconstructions. Both methods are applied under the framework of explicit time discretization and are tested solely in inviscid scenarios. We will present numerous numerical tests and provide a comparative analysis of the outcomes derived from these two computational approaches.

Comparison of the high-order Runge-Kutta discontinuous Galerkin method and gas-kinetic scheme for inviscid compressible flow simulations

TL;DR

This study benchmarks the high-order RKDG method and the Gas-Kinetic Scheme (GKS) for inviscid compressible flows, highlighting the trade-offs between RKDG’s high accuracy and sensitivity to limiters and GKS’s relaxed CFL constraints and physically grounded fluxes. By comparing across 1D and 2D Euler tests with both non-compact and compact reconstructions, the work shows that CGKS can achieve competitive or superior accuracy with fewer degrees of freedom in smooth regions and can handle shocks robustly, while RKDG maintains superior performance in certain discontinuous regimes. The results motivate a hybrid - approach that blends DG and CGKS for unified treatment of smooth and sharp features, with future extensions to viscous flows and limiter robustness. Overall, the paper provides a detailed, fair comparison and practical insights into leveraging the strengths of both high-order methods for compressible gas dynamics.

Abstract

The Runge--Kutta discontinuous Galerkin (RKDG) method is a high-order technique for addressing hyperbolic conservation laws, which has been refined over recent decades and is effective in handling shock discontinuities. Despite its advancements, the RKDG method faces challenges, such as stringent constraints on the explicit time-step size and reduced robustness when dealing with strong discontinuities. On the other hand, the Gas-Kinetic Scheme (GKS) based on a high-order gas evolution model also delivers significant accuracy and stability in solving hyperbolic conservation laws through refined spatial and temporal discretizations. Unlike RKDG, GKS allows for more flexible CFL number constraints and features an advanced flow evolution mechanism at cell interfaces. Additionally, GKS' compact spatial reconstruction enhances the accuracy of the method and its ability to capture stable strong discontinuities effectively. In this study, we conduct a thorough examination of the RKDG method using various numerical fluxes and the GKS method employing both compact and non-compact spatial reconstructions. Both methods are applied under the framework of explicit time discretization and are tested solely in inviscid scenarios. We will present numerous numerical tests and provide a comparative analysis of the outcomes derived from these two computational approaches.
Paper Structure (13 sections, 52 equations, 10 figures, 7 tables)

This paper contains 13 sections, 52 equations, 10 figures, 7 tables.

Table of Contents

  1. Introduction
  2. A brief review of the RKDG method
  3. Implementation of limiters
  4. A brief review of the gas-kinetic method
  5. Gas-kinetic evolution model
  6. Implementation of the spatial reconstruction
  7. Two-stage fourth-order temporal discretization
  8. Numerical examples
  9. One dimensional accuracy tests
  10. One dimensional problems with discontinuities
  11. Two dimensinal accuracy test
  12. Double Mach reflection problem
  13. Conclusion

Figures (10)

  • Figure 1: Stencils for one dimensional compact gas-kinetic scheme reconstruction.
  • Figure 2: Stencils for two-dimensional compact gas-kinetic scheme reconstruction.
  • Figure 3: Efficiency of simulation for smooth regions: comparisons between RKDG-$P^2$ method and third-order GKS method. The left one is based on $L^{1}$ error and the right one is based on $L^{2}$ error.
  • Figure 4: Efficiency of simulation for smooth regions: comparisons between RKDG-$P^4$ and fifth-order GKS and CGKS. The left one is based on $L^{1}$ error and the right one is based on $L^{2}$ error.
  • Figure 5: Shu-Osher problem: the density distributions and local enlargement at $t=1.8s$ with $400$ cells. Left to right: RKDG-$P^{2}$ with third-order GKS, RKDG-$P^{4}$ with fifth-order GKS, RKDG-$P^{4}$ with fifth-order CGKS.
  • ...and 5 more figures