Methods and Applications of Analysis

Volume 25 (2018)

Number 2

Conservation laws and error estimates of several classical finite difference schemes for the nonlinear Schrödinger/Gross–Pitaevskii equation

Pages: 97 – 116

DOI: https://dx.doi.org/10.4310/MAA.2018.v25.n2.a2

Authors

Ting-Chun Wang (School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, China)

Wen Zhang (School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, China)

Chen-Yi Zhu (School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, China)

Abstract

In this paper, several classical implicit finite difference schemes for solving the nonlinear Schrödinger/Gross Pitaevskii (NLS/GP) equation are revisited and analyzed. By introducing a kind of energy functionals, these schemes are proved to preserve the total energy in the discrete sense. Besides the standard energy method, a ‘cut-off’ technique and a ‘lifting’ technique are adopted to establish the optimal point-wise error estimates without any restriction on the grid ratios. Numerical results are reported to verify the theoretical analysis.

Keywords

NLS/GP equation, finite difference scheme, energy conservation, unconditional convergence, error estimate

2010 Mathematics Subject Classification

65M06, 65M12

Full Text (PDF format)

This work is supported by the National Natural Science Foundation (Grant No. 11571181), the Natural Science Foundation of Jiangsu Province (Grant No. BK20171454), and the Oing Lan Project.

Received 8 July 2016

Accepted 4 September 2018

Published 3 January 2019