Progress in gas-kinetic upwind schemes for the solution of Euler/Navier–Stokes equations – I: Overview

Three gas-kinetic upwind schemes for the solution of the Euler/Navier–Stokes equations are reviewed. They are Kinetic Flux-Vector Splitting (KFVS), Kinetic Wave/Particle Splitting (KWPS), and Bhatnagar–Gross–Krook (BGK) methods. For the Euler equations, the most sophisticated BGK scheme can be interpreted as a relaxation scheme between the two KFVS schemes with different moments, KFVS and KFVS_u0. It improves the accuracy over the KFVS scheme and the robustness over the KFVS_u0 scheme. The direct generalization of this relaxation approach to the Navier–Stokes equations leads to a much simpler BGK scheme than the one in the literature. In this simplified BGK scheme, there exist two types of particle collision time. The one in the BGK model acts as a relaxation parameter. Its role is to add some numerical dissipation from the KFVS scheme to the KFVS_u0 scheme. On the other hand, the one in the Chapman–Enskog expansion of the gas distribution function is related to physical dissipation. Following the same approach, another type of BGK schemes is further developed, which is a relaxation scheme between KWPS and KWPS_u0. In spite of the fact that the KWPS scheme is more diffusive than the KFVS scheme, a BGK scheme based on KWPS and KWPS_u0 is found not only computationally more efficient but also less diffusive than a BGK scheme based on KFVS and KFVS_u0. However, this issue needs further and more rigorous investigation by performing the numerical analysis of a model 1-D convection–diffusion equation.

► Simplest BGK scheme for Euler equations is interpreted as a relaxation scheme of two KFVS schemes.
► Extension of this relaxation approach to Navier–Stokes equations leads to a simplified BGK scheme.
► A new type of BGK schemes, a relaxation scheme of two KWPS schemes, is developed.
► BGK-KWPS is not only computationally more efficient but also less diffusive than BGK-KFVS.