IMPROVEMENT ON STABILITY AND CONVERGENCE OF A. D. I. SCHEMES

作     者：程爱杰 

作者机构：Department of Mathematics Shandong University Jinan 250100 China 

出 版 物：《Applied Mathematics and Mechanics(English Edition)》 (应用数学和力学（英文版）)

年 卷 期：1999年第20卷第1期

页      面：76-83页

核心收录：

学科分类：07[理学] 0805[工学-材料科学与工程（可授工学、理学学位）] 0802[工学-机械工程] 070102[理学-计算数学] 0701[理学-数学] 0801[工学-力学（可授工学、理学学位）] 

基　　金：National Natural Science Foundation of China NSFC 

主　　题：P-R scheme Douglas scheme parabolic partial differential equation variable coefficient H-1 energy estimating method stability and convergence 

摘      要：Alternating direction implicit (A.D.I.) schemes have been proved valuable in the approximation of the solutions of parabolic partial differential equations in multi-dimensional space. Consider equations in the form partial derivative u/partial derivative t - partial derivative/partial derivative x(a(x,y,t) partial derivative u/partial derivative x) - partial derivative/partial derivative y(b(x,y,t) partial derivative u partial derivative y) = f Two A.D.I. schemes, Peaceman-Rachford scheme and Douglas scheme will be studied. In the literature, stability and convergence have been analysed with Fourier Method, which cannot be extended beyond the model problem with constant coefficients. Additionally, L-2 energy method has been introduced to analyse the case of non-constant coefficients, however, the conclusions are too weak and incomplete because of the so-called equivalence between L-2 norm and H-1 semi-norm . In this paper, we try to improve these conclusions by H-1 energy estimating method. The principal results are that both of the two A.D.I. schemes are absolutely stable and converge to the exact solution with error estimations O(Delta t(2) + h(2)) in discrete H-1 norm. This implies essential improvement of existing conclusions.