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

In this paper, a fourth-order compact and energy conservative difference scheme is proposed for solving the two-dimensional nonlinear Schrödinger 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 + τ 2 ) in the discrete L 2 -norm with time step τ 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.