Analysis of a Finite Volume Method for a Cross-Diffusion Model in Population Dynamics

Abstract

Abstract. 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.

Mathematical analysis and numerical simulation of pattern formation under cross-diffusion

“…We have experienced that the treatment of the diffusive terms using instead the following choice [34] (imposed originally to justify the non-negativity of the approximate solutions, and coercivity of the cross-diffusion matrix)…”

“…The well-posedness of the scheme, positivity-preserving property, and convergence are addressed in detail in [34].…”

“…More suitable discretization strategies from the viewpoint of numerical analysis are given by finite elements (see for instance [33] for a related problem), and finite volume schemes (see e.g. [34][35][36]). We adopt the recent finite volume method proposed by Andreianov et al [34] for the numerical treatment of the underlying reaction-diffusion system with cross-diffusion.…”

“…[34][35][36]). We adopt the recent finite volume method proposed by Andreianov et al [34] for the numerical treatment of the underlying reaction-diffusion system with cross-diffusion. By using the linear stability analysis and the finite volume method in [34], we consider a two-species Lotka-Volterra reaction-diffusion competition planktonic system, where each species has an allelopathic effect on the other one, and we show that the cross-diffusion gives rise to the formation of patterns.…”

Convergence of approximate solutions to an elliptic–parabolic equation without the structure condition

“…In practice, this can be done via the discrete Kruzhkov lemma (see e.g. [6]). Another way to proceed is the standard multiplication technique which goes back to [2] (see e.g.…”