A New Riemann Solver And Related Issues

This paper consists of two parts. In the first part, the Riemann solvers in Godunov-type schemes are considered in the hyperbolic conservation laws systems. Based on the two-step splitting method, a new approximate Riemann solver is presented. In the first step we add a linear advection term to the equations of hyperbolic conservation laws to make the Godunov numerical flux in the interface between cells is computed easily, then the added linear advection term is thrown off in the second step. During the computing procedure, the algorithm does not need iterative technique and characteristic wave decomposition. After the new solver is expressed a form of a new numerical flux, the new scheme is proved monotonic and TVD and satisfies the entropy condition. A lot of numerical results show its robustness and the solver can be applied widely.In the second part, we show reasons that the sonic point glitch in the computation of conservation law happens. A new method is presented to cure the sonic glitch. The method that based on the two-step splitting method is similar to the present solver presented in the first part. Applying this method to many schemes with sonic point glitch, the sonic glitch is cured.