Gas Evolution Dynamics in Godunov-Type Schemes

As a continuous eeort to understand the Godunov-type schemes, following the paper \Projection in this paper we are going to study the gas evolution dynamics of the exact and approximate Riemann solvers. More speciically, the underlying dynamics of Flux Vector Splitting (FVS) and Flux Diierence Splitting (FDS) schemes will be analyzed. Since the FVS scheme and the Kinetic Flux Vector Splitting (KFVS) scheme have the same physical mechanism and numerical formulations, based on the governing equation of the discretized KFVS scheme, the weakness and advantages of FVS scheme are clearly observed. Also, in this paper, the implicit equilibrium assumption in the Godunov ux will be analyzed. Due to the numerical shock thickness related to the cell size, the numerical scheme should be able to capture both equilibrium and non-equilibrium ow behavior in smooth and discontinuous regions. The Godunov ux basically lacks the mechanism to capture nonequilibrium eeects in the artiicially enlarged numerical shock region and to stabilize the numerical shock structure. Consequently, the Godunov method is exposed to the possible spurious solutions, such as the carbuncle phenomena and odd-even decoupling, once the dissipation provided in the projection stage is not enough.