非线性椭圆问题的紧致差分格式及瀑布两网格法  

Compact Finite Difference Scheme and Cascadic Two Grid Method for Nonlinear Elliptic Problem

作　　者：李明[1] 赵金娥[1] 

机构地区：[1]红河学院数学学院,云南蒙自661199

出　　处：《重庆师范大学学报（自然科学版）》2015年第3期88-94,共7页Journal of Chongqing Normal University:Natural Science

基　　金：国家自然科学基金(No.11161014;No.11461023;No.11471166);云南省科研项目(No.2012FD054;No.2013FZ118;No.2014Y461)

摘　　要：为了讨论来源于科学工程问题的二维非线性椭圆问题的离散格式及其数值解法。首先,将泊松方程的四阶紧致差分格式推广至二维非线性椭圆问题,提出了紧致差分(CFD)格式,基于CFD格式,选取合适的步长,形成粗网格层和细网格层。在粗网格层上,使用牛顿法求得对应的非线性方程的高精度数值解;在细网格层上,运用插值算子将粗网格上的数值解进行插值,得到细层上较好的初始值,并再次使用牛顿法进行求解,提出了CFD格式下的瀑布两网格(CTG)法。数值实验表明:提出的CFD格式具有四阶计算精度,CTG法迭代步数少、计算时间短。Many science engineering problems can be attributed to solve a nonlinear elliptic problem. In this paper, a compact finite difference （CFD） scheme and a numerical method are discussed for the two dimensional nonlinear elliptic problem. Firstly, a com- pact finite difference （CFD） scheme is proposed, by combining a fourth order compact finite difference scheme of Poisson problem. Secondly, a coarse grid and a fine grid can be obtained, by choosing proper step length. On the coarse grid, Newton method is used to solve the nonlinear equation, and obtain the high accuracy approximate solution. A better initial value is provided on fine grid, by taking interpolation operator. Newton method is applied again for the initial value. Then, a cascadic two grid （CTG） method is designed. Numerical experiments of the CFD scheme and CTG method are given, which show that the new CFD scheme with accuracies of fourth order, and the CTG method is effective.

关 键 词：非线性椭圆问题 紧致差分格式 瀑布两网格法 

分 类 号：O241.6[理学—计算数学]