High-Order Spectral Difference Gas-Kinetic Schemes for Euler and Navier-Stokes Equations

High-order spectral difference gas-kinetic schemes (SDGKS) are developed for inviscid and viscous flows on unstructured quadrilateral meshes. Rather than the traditional Riemann solver, the spectral difference method is coupled with the gas-kinetic solver, which provides a time-accurate flux function at the cell interface. With the time derivative of the flux function, a two-stage fourth-order time-stepping method is adopted to achieve high-order accuracy with fewer middle stages. The stability analysis for the linear advection equation shows that fourth-order spatial and temporal discretization SDGKS is stable under CFL condition. Quantitatively, the fourth-order SDGKS is around 8% more efficient than the traditional one with the Riemann solver and the strong stability preserving five-stage fourth-order Runge-Kutta method. Both steady and unsteady tests obtained by SDGKS compare well with analytic solutions and reference results.