An efficient split-step and implicit pure mesh-free method for the 2D/3D nonlinear Gross-Pitaevskii equations

In this paper, a high-efficient, split-step, and implicit corrected parallel smoothed particle hydrodynamics (SS-ICPSPH) method is developed to simulate the dynamic systems of several nonlinear Schrödinger/Gross-Pitaevskii equations (NLSE/GPE). The proposed method is motivated by the split-step for the equation, the corrected symmetric kernel gradient for the traditional SPH and the implicit scheme for time, respectively. Meanwhile, the MPI parallel technique is introduced to enhance the computational efficiency. Firstly, the numerical accuracy and the merits of the proposed method are tested by solving 2D NLSE, and compared with the analytical results. Secondly, the new method is extended to simulate the 2D/3D two-component GPE, compared with high accuracy finite difference results. Thirdly, the proposed method is extended to investigate the sheet-like vortices in rotating Bose-Einstein condensate. Finally, the implicit corrected SPH scheme is tentatively extended to capture the propagation process of free surface wave in a rectangular pool with initial perturbation. All the numerical results show the ability and the reliability of the proposed method.