Numerical simulation of 3D nonlinear Schrödinger equations by using the localized method of approximate particular solutions. (May 2017)

• Record Type: Journal Article
• Authors: Lin, Ji; Hong, Yongxing; Kuo, Lei-Hsin; Liu, Chein-Shan
• Is Part Of: Engineering analysis with boundary elements. Volume 78(2017)
• Page Start: 20
• Page End: 25
• Publication Date: 2017-05
• Subjects: 3D nonlinear Schrödinger equations -- Localized method of approximate particular solutions -- Implicit-Euler scheme -- Closed form particular solutions; Boundary element methods -- Periodicals; Engineering mathematics -- Periodicals; Équations intégrales de frontière, Méthodes des -- Périodiques; Mathématiques de l'ingénieur -- Périodiques; Boundary element methods; Engineering mathematics; Periodicals; 620.00151
• DOI: 10.1016/j.enganabound.2017.02.002
• Languages: English
• ISSNs: 0955-7997

Abstract

Abstract: In this paper, we describe a novel sparse meshless approach to the simulations of three-dimensional time-dependent nonlinear Schrödinger equations. Our procedure is implemented in two successive steps. In the first step, the implicit-Euler scheme is applied for approximating the functional dependence of the solution on the temporal variables. Then, in the second step, the novel localized method of approximate particular solutions (LMAPS) is utilized for highly accurate and efficient numerical approximations of spatial systems. In the implementation of the LMAPS, the closed form particular solutions for the Laplace operator using the Gaussian radial basis function are used. Numerical experiments are provided to verify the stability and efficiency of this method. In summary, the proposed algorithm is efficient and stable, and the magnitude of the error is at about 10 −3 for 3D nonlinear Schrödinger problems.