A high-order multidimensional gas-kinetic scheme for hydrodynamic equations

Abstract

This paper concerns the development of high-order multidimensional gas kinetic schemes for the Navier-Stokes solutions. In the current approach, the state-of-the-art WENO-type initial reconstruction and the gas-kinetic evolution model are used in the construction of the scheme. In order to distinguish the physical and numerical requirements to recover a physical solution in a discretized space, two particle collision times will be used in the current high-order gas-kinetic scheme (GKS). Different from the low order gas dynamic model of the Riemann solution in the Godunov type schemes, the current method is based on a high-order multidimensional gas evolution model, where the space and time variation of a gas distribution function along a cell interface from an initial piecewise discontinuous polynomial is fully used in the flux evaluation. The high-order flux function becomes a unification of the upwind and central difference schemes. The current study demonstrates that both the high-order initial reconstruction and high-order gas evolution model are important in the design of a high-order numerical scheme. Especially, for a compact method, the use of a high-order local evolution solution in both space and time may become even more important, because a short stencil and local low order dynamic evolution model, i.e., the Riemann solution, are contradictory, where valid mechanism for the update of additional degrees of freedom becomes limited.