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An Efficient Meshless Method for Solving Multi-dimensional Nonlinear Schrödinger Equation

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Abstract

In this article, we study an efficient combination of the meshless local Petrov–Galerkin and time-splitting methods for the numerical solution of nonlinear Schrödinger equation in two and three dimensions. The Strang splitting technique is used to separate the original equation in two parts, linear and nonlinear. The linear part is approximated with the meshless local Petrov–Galerkin method in the space variable and the Crank–Nicolson method in time. Also, the nonlinear part can be solved analytically. We use the moving Kriging interpolation instated of the moving least squares approximation to make the shape functions of the meshless local Petrov–Galerkin method which have the Kronecker delta property, so the Dirichlet boundary condition is imposed directly and easily. In the meshless local Petrov–Galerkin method, the Heaviside step function is chosen as the test function in each sub-domain. Several test problems for two and three dimensions are presented, and the results are compared to their analytical and other numerical methods to illustrate the accuracy and capability of this technique.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Basic Sciences, Shiraz University of Technology, Shiraz, Iran

    Ali Habibirad, Esmail Hesameddini & Ameneh Taleei

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  1. Ali Habibirad
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  2. Esmail Hesameddini
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  3. Ameneh Taleei
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Correspondence to Esmail Hesameddini.

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Habibirad, A., Hesameddini, E. & Taleei, A. An Efficient Meshless Method for Solving Multi-dimensional Nonlinear Schrödinger Equation. Iran J Sci Technol Trans Sci 44, 749–761 (2020). https://doi.org/10.1007/s40995-020-00864-w

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  • DOI: https://doi.org/10.1007/s40995-020-00864-w

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