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

Boris Andreianov (1) , Mostafa Bendahmane (2) , Ricardo Ruiz Baier (3)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 IMB - Institut de Mathématiques de Bordeaux
3 EPFL - Ecole Polytechnique Fédérale de Lausanne

The 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.

Mots clés: Cross-diffusion; finite volume approximation; convergence to the weak solution; pattern-formation

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Boris Andreianov, Mostafa Bendahmane, Ricardo Ruiz Baier. Analysis of a finite volume method for a cross-diffusion model in population dynamics. 2009. ⟨hal-00458737v1⟩