奇摄动对流-扩散偏微分方程的混合算法

Abstract

讨论具有一般边界层的奇摄动对流-扩散偏微分方程,这类问题会在边界层附近出现剧烈振荡现象,产生所谓的边界层函数,其解析解无法求出.本文提出混合算法,其主要思想是引入二个过渡点将区域分为粗网格区域、中等网格区域和细网格区域,在这三个网格区域我们采用等步长.在粗网格区域采用Il’in差分格式,在细网格区域采用一般差分格式,在中等网格区域采用渐近解,新方法的总体误差是O(N-1+M-1+ε).混合算法结合了渐近解、数值解和BVT法的优势,是一个实用、有效的算法.

In this paper,singularly perturbed convection-diffusion partial differential equation with regular boundary layer is considered.It is well known that this class of problem change rapidly in boundary layers and lead to boundary layer function.It's analytic solution can't be obtained.Mixed Method was presented in this paper.Two transition points were introduced so that the interval is divided into three subintervals : coarse mesh subinterval,middle mesh subinterval and fine mesh subinterval.Equidistant mesh partition was applied in each subinterval.Asymptotic solution was used in middle mesh subinterval.Il'in finite difference scheme was used in coarse mesh subinterval.General finite difference scheme was used in fine mesh subinterval.The global estimates were proved to be O(N~(-1)+M~(-1)+ε).The new method has advantage of asymptotic solution,numerical solution and BVT method.It is useful and efficiently in application.