物理学报2018，Vol.67Issue(5)：50-59,10.DOI:10.7498/aps.67.20171791
反应扩散模型在图灵斑图中的应用及数值模拟
Application of reaction diffusion model in Turing pattern and numerical simulation
摘要
Abstract
Turing proposed a model for the development of patterns found in nature in 1952. Turing instability is known as diffusion-driven instability, which states that a stable spatially homogeneous equilibrium may lose its stability due to the unequal spatial diffusion coefficients. The Gierer–Mainhardt model is an activator and inhibitor system to model the generating mechanism of biological patterns. The reaction-diffusion system is often used to describe the pattern formation model arising in biology. In this paper, the mechanism of the pattern formation of the Gierer-Meinhardt model is deduced from the reactive diffusion model. It is explained that the steady equilibrium state of the nonlinear ordinary differential equation system will be unstable after adding of the diffusion term and produce the Turing pattern. The parameters of the Turing pattern are obtained by calculating the model. There are a variety of numerical methods including finite difference method and finite element method. Compared with the finite difference method and finite element method, which have low order precision, the spectral method can achieve the convergence of the exponential order with only a small number of nodes and the discretization of the suitable orthogonal polynomials. In the present work, an efficient high-precision numerical scheme is used in the numerical simulation of the reaction-diffusion equations. In spatial discretization, we construct Chebyshev differentiation matrices based on the Chebyshev points and use these matrices to differentiate the second derivative in the reaction-diffusion equation. After the spatial discretization, we obtain the nonlinear ordinary differential equations. Since the spectral differential matrix obtained by the spectral collocation method is full and cannot use the fast solution of algebraic linear equations, we choose the compact implicit integration factor method to solve the nonlinear ordinary differential equations. By introducing a compact representation for the spectral differential matrix, the compact implicit integration factor method uses matrix exponential operations sequentially in every spatial direction. As a result, exponential matrices which are calculated and stored have small sizes, as those in the one-dimensional problem. This method decouples the exact evaluation of the linear part from the implicit treatment of the nonlinear reaction terms. We only solve a local nonlinear system at each spatial grid point. This method combines with the advantages of the spectral method and the compact implicit integration factor method, i.e., high precision, good stability, and small storage and so on. Numerical simulations show that it can have a great influence on the generation of patterns that the system control parameters take different values under otherwise identical conditions. The numerical results verify the theoretical results.关键词
反应扩散方程/Gierer-Meinhardt模型/图灵斑图/Chebyshev谱方法Key words
reaction-diffusion equation/Gierer-Meinhardt model/Turing pattern/Chebyshev spectral method引用本文复制引用
张荣培,王震,王语,韩子健..反应扩散模型在图灵斑图中的应用及数值模拟[J].物理学报,2018,67(5):50-59,10.基金项目
国家自然科学基金(批准号:61573008, 61703290)、国防科技重点实验室基金(批准号:6142A050202)和辽宁省教育厅基金(批准号:L201604)资助的课题.Project supported by the National Natural Science Foundation of China (Grant Nos. 61573008, 61703290), the Key Laboratory Fund of National Defense Science and Technology, China (Grant No. 6142A050202), and the Liaoning Provincial Department of Education Fund, China (Grant No. L201604). （批准号:61573008, 61703290）
