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

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 signi...

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Bibliographic Details
Published in:Journal of scientific computing 2020-10, Vol.85 (1), p.4, Article 4
Main Author: Benzakour Amine, M.
Format: Article
Language:English
Subjects:
Accuracy
Algorithms
Approximation
Boundary conditions
Computational efficiency
Computational Mathematics and Numerical Analysis
Computing time
Embryology
Finite volume method
Growth models
Implicit methods
Linear algebra
Linearization
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical models
Mathematics
Mathematics and Statistics
Neural networks
Nonlinear differential equations
Numerical analysis
Partial differential equations
Robustness (mathematics)
Theoretical
Training
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Summary: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.
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-020-01310-0