Time-implicit gas-kinetic scheme

•Time implicit GKS is developed with numerical Jacobian matrix and GMRES scheme.•KFVS is chosen as the preconditioner instead of FVS.•A modified GKS flux is proposed to improve the dispersion of C-N scheme.•The high efficiency of GMRES-GKS is validated compared with LU-SGS-GKS.

The construction of time-implicit gas-kinetic scheme (GKS) is investigated to enhance its efficiency in engineering applications, especially with large time scales. GKS is based on BGK equation to simulate compressible flows, which describes particles evolution with mesoscopic distribution function. To develop an implicit GKS, classical time-implicit techniques can be directly adopted, with the employment of upwind approximate factorization for Euler equations. However, the characteristic relation of BGK equation is quite different to that of macroscopic compressible Euler/Navier-Stokes equations, which may decrease the gain from implicit techniques, especially in high-speed flows. To further improve the performance, the preconditioned generalized minimum residual (GMRES) based on numerical Jacobian matrix is considered, and the kinetic flux vector splitting (KFVS) is suggested to generate the precondition matrix. The proposed new time-implicit GKS, GMRES-GKS shows great improvements in temporal accuracy, resolution and convergence in several typical high-speed benchmark tests, ranging from transonic to hypersonic, steady to unsteady and inviscid to turbulent flows.