An unconditionally stable linearized CCD-ADI method for generalized nonlinear Schrödinger equations with variable coefficients in two and three dimensions

In this paper, we propose a three-level linearly implicit combined compact difference method (CCD) together with alternating direction implicit method (ADI) for solving the generalized nonlinear Schrödinger equation (NLSE) with variable coefficients in two and three dimensions. The method is sixth-order accurate in space variable and second-order accurate in time variable. Fourier analysis shows that the method is unconditionally stable. Comparing to the nonlinear CCD-PRADI scheme for solving the 2D cubic NLSE with constant coefficients (Li et al., 2015), current method is a linear scheme which generally requires much less computational cost. Moreover, current method can handle 3D problems with variable coefficients naturally. Finally, numerical results for both 2D and 3D cases are presented to illustrate the advantages of the proposed method.