To improve the numerical accuracy and computational efficiency of solving high-dimensional nonlinear Schrödinger/Gross-Pitaevskii (NLS/GP) equation by using traditional SPH method, a high-order split-step coupled with a corrected parallel SPH (HSS-CPSPH) method is developed by applying virtual particles to the boundary. The improvements are described as follows. Firstly, the nonlinear Schrödinger equation is divided into linear derivative term and nonlinear term based on the high-order split-step method. Then, the linear derivative term is solved by extending the first-order symmetric SPH method in explicit time integration. Meanwhile, the MPI parallel technique is introduced to enhance the computational efficiency. In this work, the accuracy, convergence and the computational efficiency of the proposed method are tested by solving the NLS equations with the periodic and Dirichlet boundary conditions, and compared with the analytical solutions. Also, the singularity of solitary waves under the periodic boundary condition is accurately obtained using the proposed particle method. Subsequently, the proposed HSS-CPSPH method is used to predict the results of complex two-dimensional and three-dimensioanl GP problems which are compared with other numerical results. The phenomenon of singular sharp angle in the propagation of nonlinear solitary wave and the process of quantum vortex under Bose-Einstein condensates with external rotation are presented accurately.
Project supported by the National Natural Science Foundation of China (Grant Nos. 11501495, 51779215), the China Postdoctoral Science Foundation of China (Grant Nos. 2015M581869, 2015T80589), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20150436), the National Key Technologies Research and Development Program of the Ministry of Science and Technology of China (Grant No. 2015BAD24B02-02), the Top-notch Academic Programs Project of Jiangsu Higher Education Institutions, China (Grant No. PPZY2015B109), and the Undergraduate Research and Innovation Project of Jiangsu Province, China (Grant No. 201611117016Z).