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

Preview Full PDF 170 877
Export citation
  • Abstract

We study compact finite difference methods for the Schr ¨odinger-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 CrankNicolson compact finite difference method and the semi-implicit compact finite difference method are both of order O(h42) in discrete l2,H1 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 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 results are reported to support our error estimates of the numerical methods.

  • Keywords

  • AMS Subject Headings

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
Copy to clipboard
The citation has been copied to your clipboard