Fourth-order compact and energy conservative difference schemes for the nonlinear Schrodinger equation in two dimensions

In this paper, a fourth-order compact and energy conservative difference scheme is proposed for solving the two-dimensional nonlinear Schrodinger equation with periodic boundary condition and initial condition, and the optimal convergent rate, without any restriction on the grid ratio, at the order of O(h(4) + tau(2)) in the discrete L-2-norm with time step tau and mesh size h is obtained. Besides the standard techniques of the energy method, a new technique and some important lemmas are proposed to prove the high order convergence. In order to avoid the outer iteration in implementation, a linearized compact and energy conservative difference scheme is derived. Numerical examples are given to support the theoretical analysis. (C) 2013 Elsevier Inc. All rights reserved.