修正局部Crank-Nicolson方法对二维非定常对流扩散方程的应用  

Application of the Modified Local Crank-Nicolson Method for Solving the Non-Steady Convection-Diffusion Equation

作　　者：郭瑞[1,2] 阿布都热西提.阿布都外力 

机构地区：[1]新疆大学数学与系统科学学院,乌鲁木齐830046 [2]石河子大学师范学院数学系,石河子832003

出　　处：《工程数学学报》2012年第1期63-71,共9页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(10961024);新疆高校科研计划资助项目(XJEDU2007I02)~~

摘　　要：本文利用修正局部Crank-Nicolson方法求解二维非定常对流扩散方程.首先,将二维非定常对流扩散方程转化为二维非定常热传导方程.其次,将二维非定常热传导方程转化为常微分方程组,利用指数函数的Trotter积公式近似该常微分方程组的系数矩阵并将其分离成分块小矩阵及Crank-Nicolson法求出结果,从而推出二维非定常对流扩散方程的修正局部Crank-Nicolson方法.所提方法具有计算量少,精度较高,无条件稳定的显著优点.最后,利用数值实验验证了所提方法的有效性,实验结果表明,所提方法能够得到与真解吻合的计算结果,因而具有很好的应用价值与推广意义.The modified local Crank-Nicolson method is applied to solve the 2-D unsteady convection diffusion equation.First,the 2-D unsteady convection diffusion equation is transformed into the 2-D unsteady heat equation,and then the 2-D unsteady heat equation is further transformed into the ordinary differential equations.By applying the Trotter product formula of exponential functions to approximate the coeffcient matrix of the ordinary differential equations so conducted,the coeffcient matrix is separated into small block matrices,and the Crank-Nicolson method is used to attain results.The modified local Crank-Nicolson method of the 2-D unsteady convection diffusion equation is thus obtained.The proposed method is of fast computational speed,high computational accuracy,unconditionally stability and with explicit expression.Finally,based on a series of experimental results,it is substantiated that the proposed method can always attain the exact solutions of the against problems.Our method is thus prattically valuable and is expected to be utilized in more real applications.

关 键 词：修正局部Crank-Nicolson法 无条件稳定性 二维非定常对流扩散方程 

分 类 号：O241.82[理学—计算数学]