Compact Finite Difference Schemes With Optimal Error Estimate For The Dissipative Nonlinear Schr(?)dinger Equation

Two compact finite difference schemes for solving a dissipative nonlinear Schr(?)dinger equation are proposed in this paper. By introducing a new auxiliary function, the dissipation term is eliminated and the original equation is transformed into a system preserving the total mass and energy of the auxiliary function. Based on the original dissipative equation and the equivalent conservative system, two efficient compact finite difference schemes are proposed. The existence, stability and convergence of the numerical solution are analyzed.For the first compact finite difference scheme,by using the energy method as well as the cut-off technique, the optimal error estimate of the numerical solution is established in maximum norm. Convergence order of the numerical solution in space and time is 4th-order and 2nd-order, respectively.For the second compact finite difference scheme,the unique solvability of the numerical solution is firstly proved by using the fixed point theorem and the standard energy method. Secondly, by using the energy method and mathematical induction as well as a H1 technique,the optimal error estimate of the numerical solution is established in the maximum norm. Same as the first one, convergence order of the numerical solution in space and time is 4th-order and 2nd-order, respectively.In order to verify the correctness of the theoretical analysis, several numerical results are carried out to test the conservative or dissipative laws and the convergence order,comparison with the existing scheme is also given to show that the novel schemes are superior.