Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model

Abstract

A positivity-preserving numerical scheme for a strongly coupled cross-diffusion model for two competing species is presented, based on a semi-discretization in time. The variables are the population densities of the species. Existence of strictly positive weak solutions to the semidiscrete problem is proved. Moreover, it is shown that the semidiscrete solutions converge to a non-negative solution of the continuous system in one space dimension. The proofs are based on a symmetrization of the problem via an exponential transformation of variables and the use of an entropy functional. Subject Classification (1991): 35K55, 65N40. Mathematics

Diffusive and nondiffusive population models

“…Remarkably, the positive definiteness of A is not necessary to obtain global existence of solutions to (7)- (9). The first global existence result for the one-dimensional cross-diffusion system without any restriction on the diffusion coefficients (except positivity) was achieved by Galiano et al [49]. Their result is based on two observations which are described in the following.…”

“…For this, we need to show L ∞ bounds for w i , which cannot be deduced from the above entropy estimate. The idea of [49] was to employ another "entropy" functional,…”

“…Unfortunately, there are no L ∞ bounds for w i in time. Therefore, Galiano et al [49] have discretized the cross-diffusion system in time (by the backward Euler scheme), obtaining a sequence of elliptic equations, which are solved recursively in time. Since time is discrete, the semidiscrete population densities are strictly positive.…”

“…In the population cross-diffusion system (7)- (8), we have b(w) = (e w1 , e w2 ) which is a gradient since χ(w) = e w1 + e w2 satisfies χ ′ = b. (Here, we assumed that α 12 = α 21 = 1, which can be achieved by a rescaling [49].) The entropy becomes…”

“…The numerical experiments confirm that segregation of the species occurs for sufficiently large cross-diffusion parameters. As mentioned above, a semi-discretization in time was proposed in [49], and the convergence of the semi-discrete solutions to a continuous one was proved. Barrett and Blowey [10] presented a convergence proof for a fully discrete finite-element approximation.…”