A finite-volume scheme for a cross-diffusion model arising from interacting many-particle population systems

Abstract

A finite-volume scheme for a cross-diffusion model arising from the mean-field limit of an interacting particle system for multiple population species is studied. The existence of discrete solutions and a discrete entropy production inequality is proved. The proof is based on a weighted quadratic entropy that is not the sum of the entropies of the population species.

Finite volumes for the Stefan-Maxwell cross-diffusion system

“…Let (T m , E m , (x K ) K∈Tm ) m≥1 and (∆t m ) m≥1 be sequences of admissible discretisations of Ω and (0, T ) respectively fulfilling condition (39). Let (u m , J m ) m = (u p , J p ) 1≤p≤P T ,m m≥1 be a corresponding sequence of discrete solutions to (33), from which a sequence of approximate solutions (u Tm,∆tm , J Em,∆tm ) m≥1 is reconstructed thanks to (37)- (38). Then there exists a weak solution (u, J) to (2)-( 5)-( 6) in the sense of Definition 1.1 such that, up to a upsequence,…”

“…From the discrete solutions (u m , J m ), m ≥ 1, the existence of which being guaranteed by Theorem 2.3, we reconstruct the piecewise constant functions u Tm,∆tm ∈ L ∞ (Q T ; A) and J Em,∆tm ∈ L 2 (Q T ; V 0 ) d thanks to formulas (37) and (38). In the convergence analysis, we also need the weakly consistent piecewise constant gradient reconstruction operators ∇ Em and ∇ Em,∆tm defined for m ≥ 1 and v ∈ R Tm (48)…”

“…Convergence proofs for finite volume approximations of cross-diffusion systems have been proposed in [3,1,19,15,38,16,21,42,29]. Most of the above contributions rely on the entropy-stability of the schemes, which is exploited thanks to the so-called discrete entropy method [20].…”

“…Recalling the elementary geometrical relation dm ∆σ = m σ d σ and the definitions (38) of J Em,∆tm and ( 48)- (49), one obtains that…”

Convergence of a finite-volume scheme for a degenerate-singular cross-diffusion system for biofilms

“…Cross-diffusion systems with nonlocal (in space) terms modeling food chains and epidemics were approximated in [2,3]. The convergence of the finite-volume scheme of a degenerate cross-diffusion system arising in ion transport was shown in [9], and the existence of a finite-volume scheme for a population cross-diffusion system was proved in [18].…”