Linearized Implicit Methods Based on a Single-Layer Neural Network: Application to Keller–Segel Models

Abstract

This paper is concerned with numerical approximation of some two-dimensional Keller–Segel chemotaxis models, especially those generating pattern formations. The numerical resolution of such nonlinear parabolic–parabolic or parabolic–elliptic systems of partial differential equations consumes a significant computational time when solved with fully implicit schemes. Standard linearized semi-implicit schemes, however, require reasonable computational time, but suffer from lack of accuracy. In this work, two methods based on a single-layer neural network are developed to build linearized implicit schemes: a basic one called the each step training linearized implicit method and a more efficient one, the selected steps training linearized implicit method. The proposed schemes, which make use also of a spatial finite volume method with a hybrid difference scheme approximation for convection–diffusion fluxes, are first derived for a chemotaxis system arising in embryology. The convergence of the numerical solutions to a corresponding weak solution of the studied system is established. Then the proposed methods are applied to a number of chemotaxis models, and several numerical tests are performed to illustrate their accuracy, efficiency and robustness. Generalization of the developed methods to other nonlinear partial differential equations is straightforward.