Management of discontinuous reconstruction in kinetic schemes

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.