@Article{CiCP-13-1357,
author = {Yong Zhang},
title = {Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System},
journal = {Communications in Computational Physics},
year = {2013},
volume = {13},
number = {5},
pages = {1357--1388},
abstract = {<p style="text-align: justify;">We study compact finite difference methods for the Schrödinger-Poisson
equation in a bounded domain and establish their optimal error estimates under proper
regularity assumptions on wave function $ψ$ and external potential $V(x)$. The Crank-Nicolson compact finite difference method and the semi-implicit compact finite difference method are both of order $\mathcal{O}$($h^4$+$τ^2$) 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.<br/></p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.251011.270412a},
url = {http://global-sci.org/intro/article_detail/cicp/7278.html}
}