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Analysis of a finite volume method for a cross-diffusion model in population dynamics

hal.structure.identifierLaboratoire de Mathématiques et Physique Théorique [LMPT]
hal.structure.identifierLaboratoire de Mathématiques de Besançon (UMR 6623) [LMB]
dc.contributor.authorANDREIANOV, Boris
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorBENDAHMANE, Mostafa
hal.structure.identifierEcole Polytechnique Fédérale de Lausanne [EPFL]
dc.contributor.authorRUIZ BAIER, Ricardo
dc.date.accessioned2024-04-04T02:29:22Z
dc.date.available2024-04-04T02:29:22Z
dc.date.created2009-10-28
dc.date.issued2011
dc.identifier.issn0218-2025
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/190169
dc.description.abstractEnThe main goal of this work is to propose a convergent finite volume method for a reaction-diffusion system with cross-diffusion. First, we sketch an existence proof for a class of cross-diffusion systems. Then standard two-point finite volume fluxes are used in combination with a nonlinear positivity-preserving approximation of the cross-diffusion coefficients. Existence and uniqueness of the approximate solution are addressed, and it is also shown that the scheme converges to the corresponding weak solution for the studied model. Furthermore, we provide a stability analysis to study pattern-formation phenomena, and we perform two-dimensional numerical examples which exhibit formation of nonuniform spatial patterns. From the simulations it is also found that experimental rates of convergence are slightly below second order. The convergence proof uses two ingredients of interest for various applications, namely the discrete Sobolev embedding inequalities with general boundary conditions and a space-time $L^1$ compactness argument that mimics the compactness lemma due to S.N.~Kruzhkov. The proofs of these results are given in the Appendix.
dc.language.isoen
dc.publisherWorld Scientific Publishing
dc.rights.urihttp://creativecommons.org/licenses/by-nc/
dc.subject.enCross-diffusion
dc.subject.enfinite volume approximation
dc.subject.enconvergence to the weak solution
dc.subject.enpattern-formation
dc.title.enAnalysis of a finite volume method for a cross-diffusion model in population dynamics
dc.typeArticle de revue
dc.identifier.doi10.1142/S0218202511005064
dc.subject.halMathématiques [math]/Analyse numérique [math.NA]
dc.subject.halMathématiques [math]/Equations aux dérivées partielles [math.AP]
dc.subject.halInformatique [cs]/Analyse numérique [cs.NA]
bordeaux.journalMathematical Models and Methods in Applied Sciences
bordeaux.page307-344
bordeaux.volume21
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue02
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-00458737
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-00458737v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Mathematical%20Models%20and%20Methods%20in%20Applied%20Sciences&rft.date=2011&rft.volume=21&rft.issue=02&rft.spage=307-344&rft.epage=307-344&rft.eissn=0218-2025&rft.issn=0218-2025&rft.au=ANDREIANOV,%20Boris&BENDAHMANE,%20Mostafa&RUIZ%20BAIER,%20Ricardo&rft.genre=article


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