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

Published online by Cambridge University Press:  03 June 2015

Yong Zhang
Yong Zhang*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P.R. China
*
*Corresponding author.Email:sunny5zhang@gmail.com
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Abstract

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 function and external potential V(x). The Crank-Nicolson compact finite difference method and the semi-implicit compact finite difference method are both of order Ҩ(h4 + τ2) in discrete l2,H1 and l∞ norms with mesh size h and time step t. For the errors ofcompact finite difference approximation to the second derivative andPoisson potential are nonlocal, thus besides the standard energy method and mathematical induction method, the key technique in analysisis to estimate the nonlocal approximation errors in discrete l∞ and H1 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 re-sults are reported to support our error estimates of the numerical methods.

Keywords

35Q5565M0665M1265M2281-08Schrodinger-Poisson systemCrank-Nicolson schemesemi-implicit schemecompact finite difference methodGronwall inequalitythe maximum principle

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Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 5 , May 2013, pp. 1357 - 1388
Copyright
Copyright © Global Science Press Limited 2013

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