Approximate-State Riemann Solvers
The method of Godunov  and its high—order extensions require the solution of the Riemann problem. In a practical computation this is solved billions of times, making the Riemann problem solution process the single most demanding task in the numerical method. In Chap. 4 we provided exact Riemann solvers for the Euler equations for ideal and covolume gases. An iterative procedure is always involved and the associated computational effort may not always be justified. This effort may increase dramatically by equations of state of complicated algebraic form or by the complexity of the particular system of equations being solved, or both. Approximate, non—iterative solutions have the potential to provide the necessary items of information for numerical purposes. There are essentially two ways of extracting approximate information from the solution of the Riemann problem to be used in Godunov—type methods: one approach is to find an approximation to the numerical flux employed in the numerical method, directly, see Chaps. 10, 11 and 12; the other approach is to find an approximation to a state and then evaluate the physical flux function at this state. It is the latter route the one we follow in this chapter.
KeywordsEuler Equation Rarefaction Wave Riemann Problem Contact Discontinuity Riemann Solver
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