We use a geometric approach, similar to van Leer’s MUSCL schemes, to construct a second-order accurate generalization of Godunov’s method for solving scalar conservation laws. By making suitable approximations we obtain a scheme which is easy to implement and total variation diminishing. We also investigate the entropy condition from the standpoint of the spreading of rarefaction waves. For Godunov’s method we obtain quantitative information on the rate of spreading which explains the kinks in rarefaction waves often observed at the sonic point.
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Goodman et al. (1988) studied this question.
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