Unconditional and optimal H1 error estimate of a Crank-Nicolson finite difference scheme for the Gross-Pitaevskii equation with an angular momentum rotation term

This paper is concerned with the time-step condition (or grid ratio condition) of finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. An optimal error estimate of the finite difference method in H1 norm is established without any constraints of the grid ratios. Besides the standard energy method, a 'cut-off' function technique and a 'lifting' technique is introduced in analyzing the proposed scheme. The analysis method used in this paper can be applied to many other finite difference schemes for solving the nonlinear Schrödinger-type equations for which previous works often require certain restriction on the grid ratios. Numerical results are reported to verify the error estimates and conservation laws.