反应扩散方程的紧交替方向差分格式  

A COMPACT ALTERNATE DIRECT IMPLICIT DIFFERENCE METHOD FOR THE REACTION DIFFUSION EQUATIONS

作　　者：孙志忠[1] 李雪玲[1] 

机构地区：[1]东南大学应用数学系,南京210096

出　　处：《计算数学》2005年第2期209-224,共16页Mathematica Numerica Sinica

基　　金：东南大学科研基金(项目号:XJ0307113)国家自然科学基金(项目号:10471023)资助项目.

摘　　要：本文研究二维常系数反应扩散方程的紧交替方向隐式差分格式.首先综合应用降阶法和降维法导出了紧差分格式,并给出了差分格式截断误差的表达式.其次引进过渡层变量,给出了紧交替方向隐式差分格式算法.接着用能量分析方法给出了紧交替方向隐式差分格式的解在离散H1范数下的先验估计式,证明了差分格式的可解性、稳定性和收敛性,在离散H1范数下收敛阶为O(τ2+h4).然后将Rechardson外推法应用于紧交替方向隐式差分格式,外推一次得到具有O(τ4+h6)阶精度的近似解.最后给出了数值例子,数值结果和理论结果是吻合的.The article is devoted to a compact alternate direct implicit difference method for the two-dimensinonal constant coefficient reaction-diffusion equations. Firstly, a compact difference scheme is derived by the combinatioin of the method of reduction of order and the method of reduction of dimension. The expression of the truncation error is given in detail. Secondly, a compact ADI difference scheme is presented by introducing a variable of intermediate value. Thirdly, a priori estimate of the solution of the compact ADI difference scheme in a discrete H1 norm is proved by the energy analysis method, with which the solvability and stability and convergence are achieved. The convergence order is O(τ2 +h4) in a discrete H1 norm. Fourthly, Richardson's extrapolation method is successfully applied to the compact ADI difference scheme and the approximate solution with accuracy O(τ4 +h6) is gained with once extrapolation. Finally, a numerical example demonstrates the theoretical results.

关 键 词：反应扩散方程 交替方向 隐式差分格式 能量分析方法 综合应用 截断误差 数值结果 降维法 降阶法 常系数 过渡层 表达式 估计式 可解性 收敛性 稳定性 外推法 收敛阶 近似解 范数 离散 

分 类 号：O241[理学—计算数学]