一类变系数椭圆型Dirichlet边值问题的差分外推格式  

Differential Extrapolation Scheme for a Class of Elliptic Dirichlet Boundary Value Problems with Variable Coefficients

作　　者：沈欣 石杨 杨雪花 张海湘 SHEN Xin;SHI Yang;YANG Xuehua;ZHANG Haixiang(College of Science,Hunan University of Technology,Zhuzhou Hunan 412007,China)

机构地区：[1]湖南工业大学理学院,湖南株洲412007

出　　处：《湖南工业大学学报》2025年第1期79-87,共9页Journal of Hunan University of Technology

基　　金：湖南省自然科学基金资助项目(2024JJ7146,2022JJ50083)。

摘　　要：对于变系数椭圆型偏微分方程的Dirichlet边值问题,首先,应用泰勒展开建立五点差分格式,并证明差分格式解的存在唯一性;其次,应用极值原理得到差分格式解的先验估计式,进一步证明其收敛性和稳定性;再次,应用Richardson外推法,建立具有四阶精度的外推格式;最后,应用Gauss-Seidel迭代方法对算例进行求解,数值结果表明Richardson外推法极大地提高了数值解的精度。In view of the Dirichlet boundary value problem of elliptic partial differential equations with variable coefficients,Taylor expansion is firstly applied for an establishment of a five point difference scheme,thus proving the existence and uniqueness of the difference scheme solution.Secondly,a prior estimation formula for the difference scheme solution can be obtained by applying the extremum principle,with its convergence and stability further proved.Thirdly,Richardson extrapolation method is applied to establish an extrapolation format with fourth-order accuracy.Finally,the Gauss-Seidel iterative method is applied to solve the numerical example,with the numerical results showing that the Richardson extrapolation method greatly improves the accuracy of the numerical solution.

关 键 词：计算数学 变系数 椭圆型偏微分方程 差分格式 RICHARDSON外推法 

分 类 号：O242.2[理学—计算数学]