An efficient time-splitting compact finite difference method for Gross-Pitaevskii equation

We propose an efficient time-splitting compact finite difference method for Gross-Pitaevskii equation (GPE). In our method, we solve the GPE in time with time-splitting technique and in space by the compact finite difference method. To find the numerical solution of the resulting discretized system in one-dimension (1D), two-dimensions (2D) and three-dimensions (3D), we apply the fast discrete Sine transform in 1D, 2D and 3D respectively and get an efficient solver for the discretized system in 1D, 2D and 3D, respectively. Our numerical algorithm at every time step does not need linear-algebraic-equations-solver, whose computation cost will be much higher when the spatial dimension increases. The method also has the merit that it is unconditionally stable and conservative. Moreover the method can achieve spectral-like accuracy in space when high-order compact finite difference method is applied. Extensive numerical tests for the GPE in 1D, 2D and 3D are presented to demonstrate the power and accuracy of the proposed numerical method.