A semi-implicit gas-kinetic scheme for smooth flows

In this paper, a semi-implicit gas-kinetic scheme (SIGKS) is derived for smooth flows based on the Bhatnagar-Gross-Krook (BGK) equation. As a finite-volume scheme, the evolution of the average flow variables in a control volume is under the Eulerian framework, whereas the construction of the numerical flux across the cell interface comes from the Lagrangian perspective. The adoption of the Lagrangian aspect makes the collision and the transport mechanisms intrinsically coupled together in the flux evaluation. As a result, the time step size is independent of the particle collision time and solely determined by the Courant-Friedrichs-Lewy (CFL) condition. An analysis of the reconstructed distribution function at the cell interface shows that the SIGKS can be viewed as a modified Lax-Wendroff type scheme with an additional term. Furthermore, the addition term coming from the implicitness in the reconstruction is expected to be able to enhance the numerical stability of the scheme. A number of numerical tests of smooth flows with low and moderate Mach numbers are performed to benchmark the SIGKS. The results show that the method has second-order spatial accuracy, and can give accurate numerical solutions in comparison with benchmark results. It is also demonstrated that the numerical stability of the proposed scheme is better than the original GKS for smooth flows.