解抛物型方程的一族高精度隐式差分格式  

A Family of High Accuracy Implicit Difference Schemes for Solving Parabolic Equations

作　　者：詹涌强[1] 张传林[2] 

机构地区：[1]华南理工大学广州学院计算机工程学院,广州510800 [2]暨南大学数学系,广州510632

出　　处：《应用数学和力学》2014年第7期790-797,共8页Applied Mathematics and Mechanics

基　　金：国家自然科学基金(61070165);广东省教育部产学研结合项目(2011B090400458)~~

摘　　要：构造了求解一维抛物型方程的一族高精度隐式差分格式.首先,推导了抛物型方程解的一阶偏导数在特殊节点处的一个差分近似式,利用该差分近似式和二阶中心差商近似式用待定系数法构造了一族隐式差分格式,通过选取适当的参数使格式具有高阶截断误差;然后,利用Fourier分析法证明了当r大于1/6时,差分格式是稳定的.最后,通过数值试验将差分格式的解与具有同样精度的其它差分格式的解和精确解进行了比较,并比较了差分格式与经典差分格式的计算效率.结果说明了差分格式的有效性.A family of implicit difference schemes with high accuracy for solving 1-dimensional parabolic equations were given.First,a difference approximation expression of the first order partial derivative of the solution to the parabolic equation was deduced at the special nodes; then this difference approximation expression and the second order central difference quotient approximation were used to construct a family of implicit difference schemes by the method of undetermined coefficients,and appropriate parameters were chosen to endow the schemes with high order truncation errors.In turn,the new difference schemes were proved to be stable as long as r was more than 1/6 with the Fourier analysis method.Finally,a numerical experiment was conduted on comparison of accuracy between the exact solutions,results of the new difference schemes and those of the other schemes with the same order truncation errors,as well as comparison of computational efficiency between the new schemes and the classical implicit difference schemes.The results demonstrate the high accuracy and efficiency of the presented difference schemes.

关 键 词：一维抛物型方程 隐式差分格式 截断误差 

分 类 号：O241.82[理学—计算数学]