期刊
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
卷 21, 期 2, 页码 307-344出版社
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0218202511005064
关键词
Cross-diffusion; finite volume approximation; convergence to the weak solution; pattern-formation
资金
- Chilean FONDECYT [7080187]
- European Research Council [ERC-2008-AdG 227058]
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 L-1 compactness argument that mimics the compactness lemma due to Kruzhkov. The proofs of these results are given in the Appendix.
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