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We present a second order extension of Godunov’s method for gas dynamics in Eulerian coordinates patterned after van Leer’s MUSCL scheme for gas dynamics in Lagrangian coordinates. The present method performs the Eulerian calculation in a single step by solving Riemann problems and characteristic equations for the fluxes in the Eulerian frame. We also make several modifications in the formulation of MUSCL, applicable to both this scheme and to the original Lagrangian scheme, all aimed at making a more robust and accurate scheme. We present the results of test caclulations in one and two space variables.
Phillip Colella (1985) studied this question.