Numerical approximation of one model of bacterial self-organization

Abstract

This paper presents finite difference approximations of one dimensional in space mathematical model of a bacterial self-organization. The dynamics of such nonlinear systems can lead to formation of complicated solution patterns. In this paper we show that this chemotaxisdriven instability can be connected to the ill-posed problem defined by the backward in time diffusion process. The method of lines is used to construct robust numerical approximations. At the first step we approximate spatial derivatives in the PDE by applying approximations targeted for special physical processes described by differential equations. The obtained system of ODE is split into a system describing separately fast and slow physical processes and different implicit and explicit numerical solvers are constructed for each subproblem. Results of numerical experiments are presented and convergence of finite difference schemes is investigated.