奇异摄动Fredholm方程积分初值问题的二阶移动网格方法  

The second-order moving grid method for a singularly perturbed Fredholm equation with the integral initial value condition

作　　者：罗丹 毛志 LUO Dan;MAO Zhi(School of Mathematics and Statistics,Jishou University,Jishou 416100,China)

机构地区：[1]吉首大学数学与统计学院,湖南吉首416100

出　　处：《湘潭大学学报（自然科学版）》2024年第3期45-56,共12页Journal of Xiangtan University(Natural Science Edition)

基　　金：贵州省第六届高层人才创新项目(2022-(2020)-040)。

摘　　要：为了克服奇异摄动Fredholm积分微分方程的精确解在很薄的初始层内的剧烈变化,提出了关于摄动参数在无穷范数意义下具有二阶一致收敛性的数值方法.针对积分初值定解问题,使用基函数方法构建指数拟合有限差分格式进行离散,其中积分部分采用复合梯形公式.局部截断误差估计和稳定性分析的直接结果是整体截断误差估计.基于精确解的边界估计、应用等分布原理,作为整体误差估计的推论,数值方法的二阶一致收敛性被证明.根据算法步骤实施迭代生成移动网格,移动网格通过等分数值解弧长产生,弧长由控制函数计算,控制函数取自整体误差估计的具体表达形式.两个数值算例验证了二阶一致收敛性,比较实验表明移动网格较Shishkin网格与Bakhvalov网格具有优越性.In order to overcome the rapid change of the exact solution of the singularly perturbed Fredholm integro-differential equation in very thin initial layer,a numerical method with the second order uniform convergence with respect to the perturbation parameter in the infinity norm is proposed.Such a definite problem with the integral initial condition is discretized by the exponentially fitted finite difference scheme,where the composite trapezoidal rule is applied into the integral term,with the basic function method.Directly resulting from the local truncation error estimate and the stability analysis,the global truncation error estimate,whose corollary is the second order uniform convergence derived from the boundary estimate and the equal distribution principle,is conducted.The steps of the moving mesh method,whose main idea is to equally distribute the arc length computed by the monitor function selected by the concrete form of the global error estimate,are listed.Two examples illustrate our theorical analyses and demonstrate that the moving mesh is superior to both the Shishkin mesh and the Bakhvalov mesh.

关 键 词：奇异摄动微分方程 FREDHOLM积分方程 移动网格 一致收敛 

分 类 号：O241.81[理学—计算数学]