Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrodinger-Poisson System
Yong Zhang 1*1 Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P.R. China.
Received 25 October 2011; Accepted (in revised version) 27 April 2012
Available online 8 October 2012
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 $\psi$ and external potential V(x). The Crank-Nicolson compact finite difference method and the semi-implicit compact finite difference method are both of order $O(h^4+\tau^2)$ in discrete $l^2,H^1$ and $l^\infty$ norms with mesh size h and time step $\tau$. 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 $l^\infty$ 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.AMS subject classifications: 35Q55, 65M06, 65M12, 65M22, 81-08
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Key words: Schrodinger-Poisson system, Crank-Nicolson scheme, semi-implicit scheme, compact finite difference method, Gronwall inequality, the maximum principle.
Email: email@example.com (Y. Zhang)