Conservative numerical methods for advanced model kinetic equations

Abstract

Model kinetic equations are often used to describe rarefied gas flows in the broad range of flow regimes. Accurate model kinetic equations were proposed by Shakhov [1] for monatomic gases and by Rykov [2] for diatomic gases. The main advantage of these model equations over other models reported in the literature is that the model collision integral contains expressions for heat fluxes and ensures their correct relaxation, e.g. correct Prandtl number for the monatomic gas. Developing conservative numerical methods for kinetic equations is a demanding task due to the nonlinear character and very high dimension of these equations. These methods are defined as methods from which the discrete conservation laws for the mass, momentum and energy follow. Non-conservative methods produce non-physical source terms in the conservation equations and may fail to converge to the proper solution as the Knudsen number is decreased unless the velocity grid is constantly refined. Due to the highly nonlinear character of the kinetic equation, both exact and model ones, the conservation property is very difficult to satisfy. The aim of this paper is to present a direct approach for ensuring conservation in the numerical methods for model kinetic equations, including Shakhov and Rykov models. The approach is based on the approximation of the conditions which are used in deriving these model equations. We construct a nonlinear system of equations defining the macro parameters in collision integrals. This system can be easily solved numerically. The resulting numerical methods do not require prohibitively fine meshes and ensures correct relaxation of heat fluxes in the continuum limits. We apply the proposed method to a plane hypersonic flow over a plate. The obtained results illustrate the robustness and accuracy of the method