The convergence of the GRP scheme

Abstract

This paper deals with the convergence of the second-order GRP(Generalized Riemann Problem) numerical scheme to the entropysolution for scalar conservation laws with strictly convex fluxes.The approximate profiles at each time step are linear in each cell,with possible jump discontinuities (of functional values and slopes)across cell boundaries. The basic observation is that the discretevalues produced by the scheme are exact averages of an approximate conservation law, which enables the use of propertiesof such solutions in the proof. In particular, the“total-variation" of the scheme can be controlled, using analyticproperties. In practice, the GRP code allows “sawteeth" profiles(i.e., the piecewise linear approximation is not monotone even ifthe sequences of averages is such). The “reconstruction" procedureconsidered here also allows the formation of “sawteeth" profiles,with an hypothesis of “Godunov Compatibility", which limits theslopes in cases of non-monotone profiles. The scheme is proved toconverge to a weak solution of the conservation law. In the case ofa monotone initial profile it is shown (under a further hypothesison the slopes) that the limit solution is indeed the entropysolution. The constructed solution satisfies the “finitepropagation speed", so that no rarefaction shocks can appear inintervals such that the initial function is monotone in their domainof dependence. However, the characterization of the limit solutionas the unique entropy solution, for general initial data, is stillan open problem.