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.

中文翻译：

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求解多维非线性Schrödinger方程的有效无网格方法

在本文中，我们研究了二维无网格Schrödinger方程数值解的无网格局部Petrov-Galerkin和时间分割方法的有效组合。使用Strang分裂技术将原始方程分为线性和非线性两部分。线性部分通过空间变量中的无网格局部Petrov-Galerkin方法和Crank-Nicolson方法及时进行近似。而且，非线性部分可以解析地求解。我们使用移动最小二乘法逼近的移动Kriging插值来制作具有Kroneckerδ属性的无网格局部Petrov-Galerkin方法的形状函数，因此直接且容易地施加Dirichlet边界条件。在无网格局部Petrov–Galerkin方法中，选择Heaviside阶跃函数作为每个子域中的测试函数。提出了二维和三维的几个测试问题，并将结果与​​它们的分析方法和其他数值方法进行了比较，以说明该技术的准确性和能力。