Volume 13, Issue 5
Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System

Yong Zhang

Commun. Comput. Phys., 13 (2013), pp. 1357-1388.

Published online: 2013-05

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  • Abstract

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.

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@Article{CiCP-13-1357, author = {}, 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 = {

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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.251011.270412a}, url = {http://global-sci.org/intro/article_detail/cicp/7278.html} }
TY - JOUR T1 - Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System JO - Communications in Computational Physics VL - 5 SP - 1357 EP - 1388 PY - 2013 DA - 2013/05 SN - 13 DO - http://doi.org/10.4208/cicp.251011.270412a UR - https://global-sci.org/intro/article_detail/cicp/7278.html KW - AB -

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.

Yong Zhang. (2020). Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System. Communications in Computational Physics. 13 (5). 1357-1388. doi:10.4208/cicp.251011.270412a
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