Randomized projection method for solving the three-component reaction-diffusion system for chemotaxis simulation

Abstract

In this paper, we apply a randomized approach to the numerical solution of two chemotaxis models: the two-component minimal non-local Keller–Segel model and the three-component reaction-diffusion model. Both models are described by nonlinear reaction-diffusion systems. To solve these problems, we use an iterative process over time. We transform the reaction-diffusion system into a matrix form using a finite-difference approximation of the equations of the system. The resulting systems of linear algebraic equations are solved by the randomized projection method with refinement. We test the time-iterative algorithm and the randomized projection method with refinement on the minimal non-local Keller–Segel model with a known exact solution and apply these algorithms to solve the three-component reaction-diffusion model. It is shown that the iterative refinement algorithm makes it possible to increase the accuracy of solutions to systems of linear algebraic equations and the entire system describing the chemotaxis model as a whole. Numerical solutions obtained using the randomized projection method with refinement are compared with exact solutions and values calculated using the LAPACKE dgesv subroutine, and good agreement between these values is obtained.