A Nonlinear Splitting Algorithm for Systems of Partial Differential Equations with self-Diffusion

TL;DR

This paper introduces a novel nonlinear splitting algorithm for reaction-diffusion systems with self-diffusion, addressing challenges in numerical approximation and stability analysis of biological models.

Contribution

A new nonlinear splitting algorithm specifically designed for PDEs with self-diffusion, including stability and convergence criteria.

Findings

Algorithm demonstrates stability in numerical tests

Convergence criteria are established theoretically

Numerical examples confirm theoretical predictions

Abstract

Systems of reaction-diffusion equations are commonly used in biological models of food chains. The populations and their complicated interactions present numerous challenges in theory and in numerical approximation. In particular, self-diffusion is a nonlinear term that models overcrowding of a particular species. The nonlinearity complicates attempts to construct efficient and accurate numerical approximations of the underlying systems of equations. In this paper, a new nonlinear splitting algorithm is designed for a partial differential equation that incorporates self-diffusion. We present a general model that incorporates self-diffusion and develop a numerical approximation. The numerical analysis of the approximation provides criteria for stability and convergence. Numerical examples are used to illustrate the theoretical results.