A General, Non-Iterative Riemann Solver for Godunov's Method

Godunov's method is characterized by the use of a Riemann problem solution to resolve discontinuities at the interface between cells. The major drawback of this method is the difficulty and the high cost of solving the (nonlinear) Riemann problem exactly, especially for materials with complex equations of state. This paper describes a simplified and noniterative approximate Riemann solver which is characterized by only two material-dependent parameters. For a given material, these parameters are the local speed of sound and a parameter which is directly related to the shock density ratio in the limit of strong shocks. These parameters are conveniently obtained from a linear fit to the experimental data for the shock Hugoniot in various materials. The approximate Riemann solver retains the essential quadratic nonlinearity which enables it to deal with the whole range of cases from weak sound waves to strong shocks.