Differential and Integral Equations

A multigrid method for pattern formation problems in biology

Chiachi Chiu and Hsiu-Chuan Wei

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Abstract

Mathematical models of many pattern formation problems in biology are reaction-diffusion systems. These systems are important for computer simulations of the patterns, parameter estimations as well as analysis of the biological properties. In order to solve reaction-diffusion systems efficiently, fast and stable numerical algorithms are essential for the pattern formation problems. In this paper, a fairly general reaction-diffusion system is considered. We propose a fully implicit discretization combined with a multigrid V-cycle solver for solving the reaction-diffusion system. Theorems about unconditional stability and convergence of the algorithm are given to show that the algorithm is highly stable and efficient. Numerical experiment results are given for two reaction-diffusion systems that can be used for generating animal coat markings. We also show the comparison results of the multigrid algorithm with other numerical algorithms.

Article information

Source
Differential Integral Equations, Volume 16, Number 2 (2003), 201-220.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060684

Mathematical Reviews number (MathSciNet)
MR1947092

Zentralblatt MATH identifier
1024.92002

Subjects
Primary: 92C15: Developmental biology, pattern formation
Secondary: 65M06: Finite difference methods 65M12: Stability and convergence of numerical methods 65M55: Multigrid methods; domain decomposition 81T80: Simulation and numerical modeling

Citation

Chiu, Chiachi; Wei, Hsiu-Chuan. A multigrid method for pattern formation problems in biology. Differential Integral Equations 16 (2003), no. 2, 201--220. https://projecteuclid.org/euclid.die/1356060684


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