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

Details

Serval ID
serval:BIB_FC1EB5EDBA8C
Type
Article: article from journal or magazin.
Collection
Publications
Institution
Title
Analysis of a finite volume method for a cross-diffusion model in population dynamics
Journal
Mathematical Models and Methods in Applied Sciences
Author(s)
Andreianov B., Bendahmane M., Ruiz-Baier R.
ISSN-L
0218-2025
Publication state
Published
Issued date
2011
Peer-reviewed
Oui
Volume
21
Pages
307-344
Language
english
Abstract
The main goal of this paper is to propose a convergent
finite volume method for a reactionâeuro"diffusion system with
cross-diffusion. First, we sketch an existence proof for
a class of cross-diffusion systems. Then the 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 Kruzhkov. The proofs of
these results are given in the Appendix.
Keywords
Cross-diffusion, finite volume approximation, , convergence to the weak solution, pattern-formation, , Nonlinear Cross, Spatial Segregation, Parabolic, Equations, Self-Diffusion, Predator-Prey, System, , Convergence, Approximation, Scheme, Inequalities
Create date
02/07/2013 10:54
Last modification date
20/08/2019 17:27
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