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Resonance in rarefaction and shock curves: local analysis and numerics of the continuation method. (English) Zbl 1478.35142

Summary: We describe certain crucial steps in the development of an algorithm for finding the Riemann solution to systems of conservation laws. We relax the classical hypotheses of strict hyperbolicity and genuine nonlinearity due to Lax. First, we present a procedure for continuing wave curves beyond points where characteristic speeds coincide, i.e. at wave curve points of maximal co-dimensionality. This procedure requires strict hyperbolicity on both sides of the coincidence locus. Loss of strict hyperbolicity is regularized by means of a Generalized Jordan Chain, which serves to construct a four-fold sub-manifold structure on which wave curves can be continued. Second, we analyze the loss of genuine nonlinearity. We prove a new result: the existence of composite wave curves when the composite wave traverses either the inflection locus or an anomalous part of the non-local composite wave curve. In this sense, we find conditions under which the composite field is well defined and its singularities can be removed, allowing use of our continuation method. Finally, we present numerical examples for a non-strictly hyperbolic system of conservation laws.

MSC:

35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
35C06 Self-similar solutions to PDEs
35D30 Weak solutions to PDEs
76S05 Flows in porous media; filtration; seepage
76T30 Three or more component flows
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