Abstract: In this paper, two compact finite difference schemes are proposed and analyzed for solving a dissipative nonlinear Schrödinger equation. Due to the difficulty in obtaining the a priori estimate of numerical solutions, it is hard to prove the convergence of the proposed schemes. In order to overcome the difficulty, we truncate the coefficient function of the nonlinear term into a global Lipschitz continuous function and use the standard energy method to establish the optimal error estimates of numerical solutions in terms of the maximum norm without any restriction on the grid ratio. The convergence rates are proved to be of the fourth-order in space and the second-order in time, respectively. Numerical results support the theoretical analysis and show the advantages of the proposed schemes compared with the existing ones.

Key words: dissipative nonlinear Schr?dinger equation, compact finite difference scheme, optimal point-wise error estimate