ANALYSIS OF A FINITE VOLUME METHOD FOR A CROSS-DIFFUSION MODEL IN POPULATION DYNAMICS
- ANDREIANOV, BORIS Laboratoire de Mathématiques CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France
- BENDAHMANE, MOSTAFA Institut de Mathématiques de Bordeaux UMR CNRS 5251, Université Victor Segalen Bordeaux 2, F-33076 Bordeaux Cedex, France
- RUIZ-BAIER, RICARDO Modeling and Scientific Computing CMCS-MATHICSE-SB, École Polytechnique Fédérale de Lausanne EPFL, CH-1015 Lausanne, Switzerland
- 2011-10-24
Published in:
- Mathematical Models and Methods in Applied Sciences. - World Scientific Pub Co Pte Lt. - 2011, vol. 21, no. 02, p. 307-344
English
The main goal of this paper 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 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 spacetime L1 compactness argument that mimics the compactness lemma due to Kruzhkov. The proofs of these results are given in the Appendix.
- Language
-
- English
- Open access status
- green
- Identifiers
-
- DOI 10.1142/s0218202511005064
- ISSN 0218-2025
- Persistent URL
- https://sonar.ch/global/documents/211570
Statistics
Document views: 34
File downloads:
- Full-text: 0