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Abstract: This paper presents a model which provides a smooth transition between a kinetic and a hydrodynamic domain. The idea is to use a buffer zone, in which both hydrodynamics and kinetic equations will be solved. The solution of the original kinetic equation will be recovered as the sum of the solutions of these two equations. We use an artificial connecting function which makes the equation on each domain degenerate at the end of the buffer zone, thus no boundary condition is needed at the transition point. Consequently, this model avoids the delicate issue of finding the interface condition in a typical domain decomposition method that couples a kinetic equation with hydrodynamic equations. A simple kinetic scheme is developed to discretize our model, and numerical examples are used to validate the method. [Copyright &y& Elsevier]

The present paper concerns two aspects for the Burnett equations. First, we are going to theoretically show the consistency between the traditional Chapman–Enskog expansion and the successive approximation for the BGK equation up to the super-Burnett order. Second, we will design a numerical scheme to efficiently solve the Burnett equations. The current approach is an improvement of the BGK-Burnett scheme [K. Xu, Phys. Fluids 15 (2003) 2077], where the locally constant collision frequency is considered. Based on the Burnett distribution function, a high order time accurate numerical flux is derived by different approaches. The resulting scheme is tested in the problem of normal shock wave and that of force-driven Poiseuille flow. [Copyright &y& Elsevier]

The present paper highlights the importance of management of the discontinuous reconstruction in the kinetic schemes for gasdynamic equation systems. Firstly, it is revealed by the analysis of the gas kinetic-BGK scheme [JCP 171 (2001) 289] that a continuous reconstruction created from a discontinuous one is a key to the successful kinetic schemes. When it is applied to a well-resolved region, the numerical flux that takes account of the collision effect correctly becomes Lax–Wendroff-like. When it is applied to an unresolved region, such as a shock layer, an appreciable numerical dissipation, which contributes to the suppression of spurious oscillations, is produced. Secondly, new kinetic schemes for the compressible Navier–Stokes (Euler) equations are developed by using the key. The numerical flux of one of the schemes is computed by using the splitting algorithm, where the effect of the molecular collision is directly taken into account and the undesirable error of the splitting algorithm in the case where the time step is much larger than the mean free time is avoided by a simple modification of the initial data. Although a discontinuous reconstruction is employed in the approximation of the initial data, the continuity is automatically taken into account in the dominant part of the numerical flux. The other schemes are the extensions of the Lax–Wendroff-type scheme to the case of the key reconstruction. Thirdly, the performance of these schemes is tested. It is demonstrated that they work as shock capturing schemes and yield fine boundary-layer profiles with a reasonable number of cells, such as 10 cells in the layer. [Copyright &y& Elsevier]