Two-dimensional Shallow Water Wave Equations And Euler Equations Numerical Shock Instability Analysis

When using the numerical schemes which can accurately capture discontinuity to calculate multidimensional fluid mechanics problems, we will find the perturbation near the shock wave increases dramatically. This is what we call numerical shock instability. Several attempts have been made to understand and cure the instability phenomenon on 2D Euler equations. But few literatures discussed the numerical shock instability of 2D shallow water equations. The nonlinear structure of the multidimensional shallow water equations does not include contact wave but has shear waves. The study of this simple structure provides an insight into the mechanism of shock instability.In the first part of this paper, we will discuss the numerical shock instability of 2D shallow water wave equations. By analyzing linear stability of some numerical schemes and testifying of some numerical test problems. We find the marginal stability of schemes and dissipation of nonlinear waves result in shock instability of shallow water equations. According to this, we design a hybrid method to remedy the nonphysical phenomenon only by slightly and locally modify the original schemes. The numerical experiments prove efficiency and robustness of hybrid scheme to eliminate shock instability of 2D shallow water equations.In the second part of this paper, we will apply the analysis method of 2D shallow water equations to 2D Euler equations, and devise corresponding hybrid scheme to remedy shock instability of 2D Euler equations.