Improvement of computational efficiency of circular function-based gas kinetic scheme by using Jacobian-free Newton-Krylov method

The gas-kinetic BGK scheme is a promising method for simulation of inviscid and viscous flows. Different from conventional Navier-Stokes (N-S) solvers, it evaluates the inviscid and viscous fluxes simultaneously by reconstructing the local solution of BGK Boltzmann equation at the cell interface. Due to its inherently superior dissipation property, it usually gives accurate and robust numerical results. However, a notable drawback of the BGK-type scheme is the low computational efficiency. Recently, aimed at reducing the computational effort, a circular function-based BGK (CBGK) scheme was developed. Nevertheless, it is still time consuming because the original scheme used an explicit way in the time integration as in most existing BGK schemes and the convergence speed can be slow. To improve the computational efficiency, an implicit CBGK scheme is developed in this paper by incorporating the Jacobian-free Newton-Krylov (JFNK) method into the scheme. Particularly, the generalized minimal residual (GMRES) approach is employed to iteratively solve the large linear equation system. With the help of the Jacobian-free approach, a faster convergence speed can be achieved without explicitly computing and storing the flux Jacobian, which is usually a large and sparse matrix. In order to reduce the number of GMRES iterations, the preconditioning is also adopted and the Lower-Upper symmetric Gauss-Seidel (LUSGS) scheme is employed as a preconditioner. For validation of the present CBGK-JFNK method, several two-dimensional inviscid and viscous test cases are investigated. The numerical results show that the needed computational time is significantly reduced as compared with the original explicit CBGK scheme and the CBGK-LUSGS method.