A Method to Solve the Reaction-Diffusion-Chemotaxis System

Abstract

Abstract The objective of this article is to use a recent developed scheme to simulate reaction-diffusion-chemotaxis equations. The solution gradient required for an accurate discretization is computed directly as an additional variable rather than interpolated from solution values around neighboring computational nodes. To achieve this goal, a supplementary equation and its associated control volume are introduced to retain a compact and accurate discretization. Scheme essentials are exposed by the numerical analysis on two-dimensional chemotaxis problems to reveal its formal accuracy. Due to its highly comprehensible and practical features, this formulation can be easily extended to solve problems for other two-dimensional rectangular grid systems. One- and two-dimensional problems are solved to verify its simulation accuracy and to study the possible formation of bacteria bands. We further perform the linearization technique to the reaction term to increase the stability of the current scheme. From the numerical analysis and computational results, it is found that the present formulation is a useful tool to solve reaction-diffusion-chemotaxis equations.