As a fundamental model to describe the nonlinear phenomenon of nature, the nonlinear Schrödinger (NLS) equation is widely used in kinds of fields, such as nonlinear optical medium,^[1–3] photonics,^[4] plasmas,^[5] Bose–Einstein condensates,^[6,7] quantum mechanics,^[8] electro-magnetic wave propagation,^[9] underwater acoustics,^[10–12] etc.

The Schrödinger equation has been widely explored and investigated in the last few years. The methods of numerical modeling for Schrödinger equation have been put forward by finite difference schemes,^[13] a Crank–Nicolson implicit scheme,^[14] higher-order compact scheme,^[15] radial basis functions,^[16] a fully implicit formula,^[17] meshless local Petrov–Galerkin (MLPG) method,^[18] and the dual reciprocity boundary element method (DRBEM),^[19] and so on.

As important numerical methods, meshless methods^[20,21] have been used for solving many complicated problems in science and engineering. It is important to develop various meshless methods of solving linear and nonlinear problems accurately and efficiently.^[22–24] The moving least-square (MLS) approximation was first used for data fitting.^[25] The MLS technique has been applied to the analysis of solid mechanics and often used to construct shape function in the element-free Galerkin (EFG) method.^[20] The EFG method is one of the most important meshless methods that have attracted much attention of researchers.^[20] This method does not require any element connectivity data, and does not suffer much degradation in accuracy when nodal arrangements are very irregular. The method has been successfully applied to a lot of science and engineering problems, such as elasticity, heat conduction, and crack growth. The MLS technique is derived from the traditional least squares method, which means that the least squares method is used at each computing point in the actual process of calculation procedure. Unfortunately, the algebraic equation system in the MLS method is sometimes ill-conditioned. In order to solve this problem, Cheng and Chen presented the improved moving least-square (IMLS) approximation.^[26] In the IMLS approximation, the basis function is chosen as an orthogonal function system, and the resulting final algebraic equation system is not ill-conditioned. Based on the IMLS approximation, the boundary element-free method is used for studying the elasticity, fracture, elastodynamics and potential problems.^[27–30] The improved element-free Galerkin (IEFG) method is a combination of IMLS approximation and EFG method. In the IEFG method,^[31–34] fewer nodes are selected in the entire domain than in the conventional EFG method. Hence, the IEFG method will definitely increase the computational efficiency. Furthermore, the IEFG method has greater computational accuracy with the same nodes than the EFG method. None of the shape functions of the MLS approximation and the IMLS approximation satisfies the property of Kronecker δ function. In order to overcome the disadvantage, Ren et al. presented Lancaster’s improved IMLS method.^[35] Based on the new shape function of IMLS method, Ren and Cheng presented interpolating element-free Galerkin (IEFG) method and interpolating boundary element-free method for potential and elasticity problems,^[35–38] and the corresponding mathematical theory was discussed.^[39] Cheng and Liew presented the IMLS Ritz and Kp Ritz method to solve heat conduction problem, wave problem and Sine-Gordon equation.^[40–43] Wang et al. put forward the interpolating element-free method and the interpolating boundary element-free method with nonsingular weight function.^[44–46] Based on the complex variable moving least-square (CVMLS) approximation and the complex variable reproducing kernel particle method (CVRKPM), the meshless method with complex variable has been presented. Chen et al. presented the complex variable reproducing kernel particle method (CVRKPM) for two-dimensional (2D) elastoplasticity, elastodynamics problems and the analysis of Kirchhoff plates.^[47–49] Cheng et al. presented the complex variable element-free Galerkin (CVEFG) method of investigating the elasticity, elastoplasticity, viscoelasticity and large deformation problems.^[50–53] Based on an improved CVMLS approximation, an improved complex variable element-free Galerkin method has been developed to solve the 2D elasticity problems and elastoplasticity.^[54,55] Deng et al. presented an interpolating complex variable element-free Galerkin method of solving the temperature field problems.^[56]

The improved element-free Galerkin (IEFG) method for the 2D unsteady Schrödinger equation is developed in this paper. In the IEFG method, the trial functions are approximated by the IMLS technique, the energy functional and the final algebraic equation system that is gained via the Galerkin method, the boundary conditions are imposed by penalty function. Numerical examples show that the present technique leads to good accuracy and efficiency.

In this section, three numerical examples are demonstrated to verify the efficiency and accuracy of the IEFG method. The implementation of the IEFG method is also presented. In all numerical examples, cubic spline function is chosen as a weight function, and the linear basis function is used.

3.1. Example 1

Consider Eq. (1) on Ω = [0,1] × [0,1] with

and

The analytical solution is

The IEFG method is used to solve the above equation with penalty factor α = 10⁵ and d_max = 2.3 in time steps of Δt = 0.01. Figure 1 shows the numerical and solutions for real part of u(x,y,t) at t = 0.5, 1, 2, 3, 4, and 6, respectively. Figure 2 describes the numerical solution and exact solution for imaginary part of u(x,y,t) when t = 0.5, 1, 2, 3, 4, and 6, respectively. In Figs. 3, 4, and 5, the surfaces of real part and imaginary part of u(x,y,t) are plotted when t = 3, 4, and 6, respectively. Table 1 presents the relative errors for the real part and imaginary part of the solution and the CPU time at t = 2. As can seen from Table 1, the smaller time step is chosen, the smaller relative error is obtained. Table 2 shows the relationship between the relative error and number of nodes for the IEFG method at time t = 1. Tables 1 and 2 show that the error norm decreases with the increase of the number of nodes and decrease of time step.

Fig. 1.

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	Fig. 1. Real parts of exact solution and numerical solution of u(x,y,t).

Fig. 2.

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	Fig. 2. Imaginary parts of exact solution and numerical solution of u(x,y,t).

Fig. 3.

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	Fig. 3. Surfaces of (a) real part and (b) imaginary part of u(x,y,t) at time t = 3.

Fig. 4.

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	Fig. 4. Surfaces of (a) real part and (b) imaginary part of u(x,y,t) at time t = 4.

Fig. 5.

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	Fig. 5. Surfaces of (a) real part and (b) imaginary part of u(x,y,t) at time t = 6.

Table 1.

Table 1.

Table 1. Relative errors and CPU times for the IEFG method at the time t = 2. . Δt Real part Imaginary part CPU time 0.1 8.7895 × 10−4 7.8986 × 10−4 3.5 0.01 3.1875 × 10−4 3.2848 × 10−4 7.6 0.001 2.1079 × 10−5 1.9846 × 10−5 11.4 0.0001 1.3195 × 10−5 8.5743 × 10−6 26.5

Table 1. Relative errors and CPU times for the IEFG method at the time t = 2. .

Table 2.

Table 2.

Table 2. Dependences of relative error on node number for the IEFG method at t = 1. . Number of nodes 225 144 100 70 relative error norm 5.7479 × 10−6 3.2745 × 10−5 6.3679 × 10−5 1.8794 × 10−4

Table 2. Dependences of relative error on node number for the IEFG method at t = 1. .

3.2. Example 2

Consider the case f(x,y) = 0 in Eq. (1) on Ω = [0,1] × [0,1] with

which derives the analytical solution

and the Dirichlet boundary conditions can be obtained from Eq. (52).

The IEFG method is used with penalty factor α = 10⁵ and d_max = 2.3 in time steps of Δt = 0.01.

Figure 6 shows the numerical solution and analytical solution for real part of u(x,y,t) at time t = 0.5, 1, 2, 3, 4, and 6, respectively. Figure 7 shows the numerical solution and exact solution for imaginary part of u(x,y,t) at t = 0.5, 1, 2, 3, 4, and 6, respectively. In Figs. 8–10, the surfaces of real part and imaginary part of u(x,y,t) are plotted at t = 1, 3, and 6, respectively.

Fig. 6.

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	Fig. 6. Real parts of analytical solution and numerical solution of u(x,y,t).

Fig. 7.

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	Fig. 7. Imiginary parts of analytical solution and numerical solution of u(x,y,t).

Fig. 8.

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	Fig. 8. Surfaces of (a) real part and (b) imaginary part of u(x,y,t) at time t = 1.

Fig. 9.

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	Fig. 9. Surfaces of (a) real part and (b) imaginary part of u(x,y,t) at time t = 3.

Fig. 10.

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	Fig. 10. Surfaces of (a) real part and (b) imaginary part of u(x,y.t) at time t = 6.

3.3. Example 3

Consider Eq. (1) on Ω = [0, 1] × [0, 1] with

and the analytical solution is

The initial and boundary conditions are easily derived from Eq. (54) as

The IEFG method is used with penalty factor α = 10⁵ and d_max = 2.3 in time steps of Δt = 0.01. Figure 11 shows the numerical solution and analytical solution for real part of u(x,y,t) when t = 0.5, 1, 2, 3, 4, and 6, respectively. Figure 12 describes the numerical solution and exact solution for imaginary part of u(x,y,t) when t = 0.5, 1, 2, 3, 4, and 6, respectively. The surfaces of real and imaginary part of u(x,y,t) are plotted at t = 1, 3, and 6, respectively, in Figs. 13–15.

Fig. 11.

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	Fig. 11. Real parts of analytical solution and numerical solution of u(x,y,t).

Fig. 12.

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	Fig. 12. Imaginary parts of analytical solution and numerical solution of u(x,y,t).

Fig. 13.

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	Fig. 13. Surfaces of (a) real part and (b) imaginary part of u(x,y,t) at time t = 6.

Fig. 14.

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	Fig. 14. Surfaces of (a) real part and (b) imaginary part of u(x,y,t) at time t = 4.

Fig. 15.

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	Fig. 15. Surfaces of (a) real part and (b) imaginary part of u(x,y,t) at time t = 6.

From these figures, it is evident that the present numerical results obtained by the IEFG method are in excellent agreement with the analytical solutions.

The IEFG approach to solving 2D Schrödinger equation is developed and numerically implemented in the present work. The IMLS approximation is used to approximate the 2D trial function. By using the variation method, the final algebraic equation system is obtained due to energy functional. In the IMLS approximation, the basis function is chosen as an orthogonal function system, and the resulting final equation system is no more ill-conditioned due to involving no inverse matrix. It is demonstrated that the IMLS technique has a higher efficiency and precision than the traditional MLS approximation. Owing to its simple implementation, the IEFG method has the potential to replace the difference method and the finite element method for solving 2D Schrödinger equation and any other nonlinear partial differential equation.