Numerical simulation of 3D nonlinear Schrödinger equations by using the localized method of approximate particular solutions

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.