The Riemann Solver of Osher
Osher’s approximate Riemann solver is one of the earliest in the literature. The bases of the approach were communicated in the papers by Engquist and Osher in 1981  and Osher and Solomon the following year . Applications to the Euler equations were published later in a paper by Osher and Chakravarthy . Since then the scheme has gained increasing popularity, particularly within the CFD community concerned with Steady Aerodynamics; see for example the works of Spekreijse , , Hemker and Spekreijse , Koren and Spekreijse , Qin et. al. , , , , , , . One of the attractions of Osher’s scheme is the smoothness of the numerical flux; the scheme has also been proved to be entropy satisfying and in practical computations it is seen to handle sonic flow well. A distinguishing feature of the Osher scheme is its performance near slowly-moving shock waves; see Roberts , Billett and Toro  and Arora and Roe . The scheme is closely related to the Flux Vector Splitting approach described in Chap. 8 and, as Godunov’s method of Chap. 6, it is a generalisation of the CIR scheme described in Chap. 5 for linear hyperbolic systems with constant coefficients. For a scalar conservation law, van Leer  studied in detail the relationship between the Osher scheme and some other Riemann solvers available at the time. Useful background material for reading this chapter is found in the previous Chaps. 2, 3, 5, 6, 8 and 9.
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