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.