An efficient high-order finite difference gas-kinetic scheme for the Euler and Navier–Stokes equations

•Maybe the first high-order finite-difference GKS solver for NS equation.•Direct use of the space and time dependent NS gas distribution function.•BGK flux solver based on a continuous initial flow distribution.•More efficient than same order macroscopic based finite-difference method.•Same code structure as standard finite-difference scheme. Based on the temporal evolution of the Navier-Stokes gas distribution function and Weighted Essential Non-Oscillatory (WENO) interpolation, a high-order finite difference gas-kinetic scheme (FDGKS) is constructed. Different from the previous high-order finite volume gas-kinetic method [Li, Xu, and Fu, J. Comput. Phys. vol. 229, pp. 6715 (2010)], which uses a discontinuous initial reconstruction at the cell interface, the present scheme is a finite-difference one with a continuous flow distribution at the grid point. The time-accurate solution of the gas distribution function permits the FDGKS to be a one-step high-order scheme without multi-step Runge-Kutta temporal matching, which significantly reduces the computational time. Many numerical tests in solving one and two-dimensional Euler and Navier-Stokes equations demonstrate that FDGKS is a highly stable, accurate, and efficient scheme, which captures discontinuities without oscillations.