Methods and Applications of Analysis
Volume 25 (2018)
Conservation laws and error estimates of several classical finite difference schemes for the nonlinear Schrödinger/Gross–Pitaevskii equation
Pages: 97 – 116
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.
NLS/GP equation, finite difference scheme, energy conservation, unconditional convergence, error estimate
2010 Mathematics Subject Classification
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