反应扩散方程的紧交替方向差分算法

Compact ADI difference scheme for solving reaction-diffusion equations

研究了三维常系数反应扩散方程的紧交替方向隐式差分格式.首先综合运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;利用Fouier稳定性方法证明了差分格式的稳定性和收敛性,且收敛阶为O(τ2+h4),并应用Richardson外推法外推一次得到具有O(τ4+h6)阶精度的近似解.数值实验结果证实,数值结果和理论结果是吻合的.

A compact alternate direct implicit difference method for the three-dimensinonal constant coefficient reaction-diffusion equations is studied.Firstly,a compact difference scheme is derived by the combination of the method of reduction of order and the method of reduct ionofdimension.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,the stability and convergence of compact ADI difference scheme are achieved by useing Fourier method,and the convergence order is O（τ^2 ＋ h^4）.Fourthly,Richardson′s extrapolation method is successfully applied to the compact ADI difference scheme and the approximate solution with accuracy O（τ^4 ＋ h^6） is gained with once extrapolation.Finally,a numerical example demonstrates the theoretical results.