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A condition on the numerical flux for semidiscrete approximations to scalar, nonconvex conservation laws is introduced, and shown to guarantee convergence to the correct physical solution. An equality which can be used to impose an entropy inequality for approximations to systems of equations is obtained. Roe’s scheme is modified to satisfy this inequality. These considerations also lead to a simple closed form expression for the solution to the Riemann problem for scalar, nonconvex conservation laws.
Stanley Osher (Sun,) studied this question.