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

Abstract

. In this paper, several classical implicit finite difference schemes for solving the nonlin- ear Schr¨odinger/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.