A new gas-kinetic scheme based on analytical solutions of the BGK equation

Up to an arbitrary order of the Chapman–Enskog expansion of the kinetic Bhatnagar–Gross–Krook (BGK) equation, a corresponding analytic solution can be obtained. Based on such a compact exact solution, a new gas-kinetic scheme is constructed for the compressible Navier–Stokes equations. Instead of using a discontinuous initial condition in the gas-kinetic BGK–NS method Xu [K. Xu, A gas-kinetic BGK scheme for the Navier–Stokes equations, and its connection with artificial dissipation and Godunov method, J. Comput. Phys. 171 (2001) 289–335.], the new scheme starts with a continuous initial flow distribution at a cell interface which is obtained through an upwind-biased WENO reconstruction, and uses the time accurate solution for the flux evaluation. The new kinetic scheme not only preserves favorable properties of the existing BGK–NS method, such as stability, high resolution, and good performance in capturing discontinuity, but also achieves a very high efficiency, which is even more efficient than the same order well-defined classical finite difference scheme based on the macroscopic governing equations directly. The stability, accuracy, and efficiency of the new scheme are evaluated quantitatively through numerical tests. The new scheme captures sharp discontinuity without post-shock oscillation, and has high accuracy in resolving viscous solution. Due to the use of time-accurate analytical solution, the overall performance of the new scheme is superior in comparison with the finite difference or finite volume schemes which start from the same initial reconstruction and use the numerical Runge–Kutta methods for time accuracy.