The convergence of the GRP scheme
type: Item
instance of: scholarly article
title: The convergence of the GRP scheme
zbMATH Open document ID: 1167.65050
author: Matania Ben-Artzi
author: J. Falcovitz
author: Jiequan Li
published in: Discrete and Continuous Dynamical Systems
publication date: 2009-02-19T00:00:00Z
review text: This paper deals with the convergence of the second-order GRP (Generalized Riemann Problem) numerical scheme to the entropy solution 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 discrete values produced by the scheme are exact averages of an approximate conservation law , which enables the use of properties of such solutions in the proof. In particular, the ``total-variation'' of the scheme can be controlled, using analytic properties. In practice, the GRP code allows ``sawteeth'' profiles (i.e., the piecewise linear approximation is not monotone even if the sequences of averages is such). The ``reconstruction'' procedure considered here also allows the formation of ``sawteeth'' profiles, with an hypothesis of ``Godunov Compatibility'', which limits the slopes in cases of non-monotone profiles. The scheme is proved to converge to a weak solution of the conservation law. In the case of a monotone initial profile it is shown (under a further hypothesis on the slopes) that the limit solution is indeed the entropy solution. The constructed solution satisfies the ``finite propagation speed'', so that no rarefaction shocks can appear in intervals such that the initial function is monotone in their domain of dependence. However, the characterization of the limit solution as the unique entropy solution, for general initial data, is still an open problem.
reviewed by: Yasar Sozen
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L67
zbMATH DE Number: 5509423
zbMATH Keywords: Hyperbolic conservation laws
zbMATH Keywords: GRP scheme
zbMATH Keywords: Convergence
zbMATH Keywords: TVD
zbMATH Keywords: entropy
zbMATH Keywords: generalized Riemann problem
zbMATH Keywords: Godunov compatibility
zbMATH Keywords: entropy solution
MaRDI profile type: Publication
full work available at URL: dcds.2009.23.1
OpenAlex ID: W2055839414
DOI: 10.3934/DCDS.2009.23.1
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description: scientific article; zbMATH DE number 5509423