求解极坐标系下反应扩散方程的紧致隐积分因子方法

COMPACT IMPLICIT INTEGRATION FACTOR METHOD FOR SOLVING REACTION DIFFUSION EQUATION IN POLAR COORDINATE

反应扩散方程在物理、化学和生物等领域有着重要的应用.以往的工作主要在矩形区域上考虑求解,本文研究圆形和环形区域上求解反应扩散方程.首先将反应扩散方程写成极坐标形式,利用二阶有限差分方法在空间r方向和θ方向分别进行离散.将网格上的数值解以矩阵形式表示,并且将微分算子离散成矩阵形式,从而得到紧致形式下的非线性常微分方程组,然后应用隐积分因子方法求解该非线性常微分方程组.紧致隐积分因子方法不仅降低了存储量,而且在每一个时间层只需要求解局部的非线性代数方程组.最后给出数值算例,选取带有精确解的反应扩散方程以及Schnakenberg模型,在圆形和环形区域上求解反应扩散方程组,数值结果显示该方法能够快速且准确地计算.

Reaction diffusion equations are widely used in physics,chemistry and biology.The previous work mainly considered the solution on the rectangular region.In this paper,the reaction-diffusion equations in circular and annular regions are studied.First,the reactiondiffusion equation is written in polar coordinate form,and the second-order finite difference method is used to discretize in spatial direction.The numerical solution on the grid is written in the matrix form,and the differential operator is discretized into matrix form,thus the nonlinear ordinary differential equations in compact form are obtained.Then the implicit integration factor method is used to solve the nonlinear ordinary differential equations.The compact implicit integral factor method not only reduces the storage,but also only needs to solve the local nonlinear algebraic equations at each time level.Finally,the numerical examples are given.The reaction-diffusion equation with exact solutions and Schnakenberg model are selected to solve on the circular and annular regions.The numerical results show that the computation with our method is efficient and accurate.