Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrodinger-Poisson System  

作　　者：Yong Zhang 

机构地区：[1]Department of Mathematical Sciences,Tsinghua University,Beijing,100084,P.R.China

出　　处：《Communications in Computational Physics》2013年第5期1357-1388,共32页计算物理通讯（英文）

基　　金：supported by Ministry of Education of Singapore grant R-146-000-120-112;the National Natural Science Foundation of China(Grant No.11131005);the Doctoral Programme Foundation of Institution of Higher Education of China(Grant No.20110002110064).

摘　　要：We study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave functionψand external potential V(x).The CrankNicolson compact finite difference method and the semi-implicit compact finite difference method are both of order O(h^(4)+τ^(2))in discrete l^(2),H^(1) and l^(∞) norms with mesh size h and time step τ.For the errors of compact finite difference approximation to the second derivative and Poisson potential are nonlocal,thus besides the standard energy method and mathematical induction method,the key technique in analysis is to estimate the nonlocal approximation errors in discrete l^(∞) and H^(1) norm by discrete maximum principle of elliptic equation and properties of some related matrix.Also some useful inequalities are established in this paper.Finally,extensive numerical results are reported to support our error estimates of the numerical methods.

关 键 词：Schrodinger-Poisson system Crank-Nicolson scheme semi-implicit scheme compact finite difference method Gronwall inequality the maximum principle 

分 类 号：O17[理学—数学]