Open AccessNumerical study of nonlinear Schrödinger equation with high-order split-step corrected smoothed particle hydrodynamics method
Author(s) -
Tao Jiang,
Jin-Jing Huang,
Lu Liu,
Jiaojiao Ren
Publication year - 2019
Publication title -
wuli xuebao
Language(s) - English
Resource type - Journals
ISSN - 1000-3290
DOI - 10.7498/aps.68.20190169
Subject(s) - nonlinear system , singularity , curvilinear coordinates , boundary value problem , physics , vortex , dirichlet boundary condition , convergence (economics) , boundary (topology) , mathematical analysis , schrödinger equation , mathematics , classical mechanics , mechanics , quantum mechanics , economics , economic growth
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.