Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations

The generalized Riemann problem (GRP) scheme for the Euler equations and gaskinetic scheme (GKS) for the Boltzmann equation are two high resolution shock capturing schemes for fluid simulations. The difference is that one is based on the characteristics of the inviscid Euler equations and their wave interactions, and the other is based on the particle transport and collisions. The similarity between them is that both methods can use identical MUSCL-type initial reconstructions around a cell interface, and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time-dependent flux function. Although both methods have been applied successfully to the inviscid compressible flow computations, their performances have never been compared. Since both methods use the same initial reconstruction, any difference is solely coming from different underlying mechanism in their flux evaluation. Therefore, such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances. Since GRP is so faithfully solving the inviscid Euler equations, the comparison can be also used to show the validity of solving the Euler equations itself. The numerical comparison shows that the GRP exhibits a better computational efficiency, and has comparable accuracy with GKS for the Euler solutions in 1D case, but the GKS is more robust than GRP. For the 2D high Mach number flow simulations, the GKS is absent from the shock instability and converges to the steady state solutions faster than the GRP. The GRP has carbuncle phenomena, likes a cloud hanging over exact Riemann solver. The GRP and GKS use different physical processes to describe the flow motion starting from a discontinuity. One is based on the assumption of equilibrium state with infinite number of particle collisions, and the other starts from the non-equilibrium free transport process to evolve into an equilibrium one through particle collisions. The different mechanism in the flux evaluation deviates their performance. Through this study, we may conclude scientifically that it is NOT valid to use the Euler equations to construct numerical fluxes in a discretized space. To adapt Navier-Stokes (NS) Preprint submitted to Elsevier 31 October 2010 equations is NOT valid either because the NS has no any account on the strength of the discontinuity. A direct modeling of the physical process in the discretized space is necessary in the construction of numerical scheme. This process is similar to the modeling in deriving the governing equations, but the control volume here cannot be shrunk to zero.