From kinetic theory of multicellular systems to hyperbolic tissue equations: Asymptotic limits and computing

Abstract

This paper deals with the analysis of the asymptotic limit toward the derivation of macroscopic equations for a class of equations modeling complex multicellular systems by methods of the kinetic theory. After having chosen an appropriate scaling of time and space, a Chapman–Enskog expansion is combined with a closed, by minimization, technique to derive hyperbolic models at the macroscopic level. The resulting macroscopic equations show how the macroscopic tissue behavior can be described by hyperbolic systems which seem the most natural in this context. We propose also an asymptotic-preserving well-balanced scheme for the one-dimensional hyperbolic model, in the two-dimensional case, we consider a time-splitting method between the conservative part and the source term where the conservative equation is approximated by the Lax–Friedrichs scheme.