Projection Dynamics in Godunov-type Schemes I: to the Physical Understanding of Post-shock Oscillations

There are two stages in the 1st-order Godunov-type schemes to update ow variables, the gas evolution stage for the numerical uxes across cell interface and the projection stage for the reconstruction of constant state inside each cell. Ideally, the evolution stage should be based on the exact Euler solutions, the so-called Riemann solver. In this paper, we will show that some anomalous phenomena, such as post-shock oscillations, are caused from the underlying unsteady dissipative mechanism from the projection stage. Based on physical model, we are going to analyze and evaluate quantitatively the projection dynamics and compare our theoretical analysis with numerical observations, such as the relation between oscillation amplitude and the shock speed. The conclusion is that any ow solvers based on the exact Euler solutions are not adequate to compensate the projection errors. In order to get a correct representation of ow motion in the discretized space and time, the consistent dissipative terms must be added in the ux functions and solve the Navier-Stokes-like equations directly.