A comment on the computation of non-conservative products

Journal of Computational Physics 229 (2010) 2759–2763Contents lists available at ScienceDirectJournal of Computational Physicsjournal homepage:www.elsevier.com/locate/jcpShort NoteA comment on the computation of non-conservative productsa, ,1 b,2*Rémi Abgrall , Smadar KarniaINRIA and University of Bordeaux, Team Bacchus, Institut de Mathématiques de Bordeaux, 351 Cours de la Libération, 33 405 Talence Cedex, FrancebDepartment of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USAarticle info abstractArticle history: We are interested in the solution of non-conservative hyperbolic systems, and consider inReceived 3 October 2009 particular the so-called path-conservative schemes (see e.g. [2,3]) which rely on the theo-Accepted 14 December 2009 reticalwork in[1].Theexample ofthestandard Eulerequationsfora perfectgas isusedtoAvailable online 29 December 2009illuminate some computational issues and shortcomings of this approach. 2009 Elsevier Inc. All rights reserved.Keywords:Non conservative hyperbolic systemsPath schemesDiscontinuous solutionsShock waves1. IntroductionNon-conservative hyperbolic systems arise in a wide range of applications, which makes their theoretical study andnumericalapproximationaveryimportanttopic.Weareinterestedinthenumericalsolutionofnon-conservativehyperbolicsystems@U @UþAðUÞ ¼ 0; ð1Þ@t @xpsubject to initial conditions. Here U 2XR .The challenge here is twofold: first, to generalize the notion of weak ...