Numerical Generalized Characteristic Analysis Method In Gas Dynamics

We study Riemann problems, the reflection of planar rarefaction wave for two-dimension Euler equations and Riemann problems for Chaplygin gas by using numerical generalized characteristic analysis method.In section 2, we introduce some useful concepts of hyperbolic systems firstly. The general theories about one-dimensional and two-dimensional hyperbolic systems are intro-duced in the latterly.In section 3, we discuss Riemanna problems for two-dimensional Euler equations. By studying the compatibility conditions of the initial data and numerical generalized characteristic analysis method, we find some new results:(1), There exist transonic shocks and rarefaction simple waves in the solutions to the caseJ12-J23-J34-J41- and transonic shocks and supersonic shocks to the case J12-J23+J34-J41+.(2), According to a degree of freedom in initial values, we may observe different numerical results as the initial values change, rarefaction simple wave or shock wave may occur.In section 4, We discuss the reflection of planar rarefaction wave for two-dimension Eule equations by using numerical generalized characteristic analysis method and find that there are two kinds of reflections:1. Regular reflection-like. The entire rarefaction wave strikes on and is reflected at the rigid wall. The reflected wave is a compressive simple wave (called von Neumann wave) which goes into a critical transonic shock.2. Mach reflection-like. The entire rarefaction wave is divided into two parts. The former part is the same as the above, but the tail of it is sonic. From then on, the rest part hits a sonic curve before they attach the rigid wall and is reflected at the sonic curve. The sonic curve plays the role of the triple point in Mach reflection of shock. The reflected wave is the same as the former one. More interesting, a rarefactive supersonic region appears in the neighborhood of the corner. As the strength of the rarefaction wave is larger than a critical value, the supersonic region is connected with the von Neumann wave.In section 5, we discuss the Riemanna problem for the two-dimensional Chaplygin gas. We divide the problem into seventeen cases and discuss case by case except for the case 2J+ +2J- by numerical generalized characteristic analysis method.