Thomas Chun Long Yue - A mesh-free numerical method for three-dimensional non-linear Schrodinger equation

Birkbeck, University of London

"The numerical solution of time-dependent Nonlinear Schrödinger Equation is a long standing problem encountered in many areas of Physics such as the modelling of fiber optical device, nonlinear water-waves and Bose-Einstein Condensates. Recent advancement in experimental studies in Bose-Einstein Condensates suggest the need for an efficient method to approximate the Nonlinear Schrödinger Equation with arbitrary trapping potential. In this presentation, we introduce a mesh-free numerical scheme to solve the three-dimensional (3D) time-dependent Nonlinear Schrodinger Equation using radial basis functions and asymmetric collocation. Combining the use of radial basis functions for spatial domain and generalized theta scheme for time domain, the presented method is truly mesh-free. A series of numerical experiments are conducted to examine the proposed method. The numerical results are compared with the analytical solutions given in existing literatures. Depending on the choice of radial basis function and the corresponding shape parameter, our numerical method is shown to be accurate by providing a maximum relative error of O(10e-3) and the root mean square error between O(10e-5)-O(10e-3)."

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