Abstract
A linear barycentric rational collocation method (LBRCM) for solving Schrodinger equation (SDE) is proposed. According to the barycentric interpolation method (BIM) of rational polynomial and Chebyshev polynomial, the matrix form of the collocation method (CM) that is easy to program is obtained. The convergence rate of the LBRCM for solving the Schrodinger equation is proved from the convergence rate of linear barycentric rational interpolation. Finally, a numerical example verifies the correctness of the theoretical analysis.
1. Introduction
Schrodinger equation (SDE) is widely used in atomic physics, nuclear physics and solid physics, quantum mechanics, and so on. SDE is only applicable to nonrelativistic particles with low velocity, and there is no description of particle spin. In this paper, we are concerned with solving the numerical solution of the SDE:where h is reduced Planck constant and m denotes quality. In [1], the fractional Schrodinger–Choquard equation with blow-up criteria and instability of normalized standing waves is studied. In [2], the finite-difference time-domain (FDTD) method is studied to solve SDE. In [3], nonlinear magnetic Schrodinger–Poisson type equation is studied. In [4], high-order multiscale discontinuous Galerkin method for one-dimensional stationary SDEs with oscillating solutions is presented. In [5], sixth-order nonlinear SDE is concerned by factorization formula and an analytical method. In [6], nonlinear SDEs are solved by the iterative method. In [7], the two-dimensional Klein–Gordon SDEs are solved by linear compact alternating direction implicit (CADI) scheme.
For getting the equidistant node of the barycentric formula, Floater [8–10] has proposed a reasonable interpolation method; in particular, equidistant distribution nodes and the quasi-equidistant nodes have high numerical stability and accuracy of interpolation [11, 12]. In [13, 14], the linear barycentric rational collocation method (LBRCM) have been used to solve the integro-differential equation. Wang et al. [15–17] have expanded the application fields of the collocation method (CM), such as initial value problems, plane elasticity problems, and nonlinear problems. LBRCM for solving heat conduction equation and biharmonic equation are studied in [18, 19].
In this paper, a LBRCM for solving SDE is proposed. According to the barycentric interpolation method (BIM) of rational polynomial and Chebyshev polynomial, the matrix form of the collocation method that is easy to program is obtained. The convergence rate of the LBRC method for solving the telegraph equation is proved from the convergence rate of linear barycentric rational interpolation (LBRI). Finally, a numerical example verifies the correctness of the theoretical analysis.
The remaining of this paper is planned as follows. Section 2 presents the differentiation matrices, CM for SDE, and the matrix form of CM. In Section 3, the convergence rate is proved. Finally, a numerical example verifies the theoretical analysis.
2. Differentiation Matrices of SDE
We partition the interval and into and with , and , for the uniform partition with and . For with will be the uniform partition.
Consider the barycentric interpolation function (BIF) asand its barycentric interpolation approximation iswherewhere , andwhere is the basis function, and is the value at point . Combining equations (1) and (5), we obtainand then, we haveandwhereandandand is Kronecher product of matrix. In the following, we define the Kronecher product of matrix and aswhere
3. Convergence Rate and Error Analysis
The barycentric rational interpolants of function (BRIF) with and its error convergence rate isandwhereandwhere
The following Lemma was proved by Jean-Pau Berrut in [11].
Lemma 1. (see [11]), For defined in (16), we haveFor the BRIF with , we can get the barycentric rational interpolation (BRI):whereand ,By the error term of Newton–Cotes rule for two-dimensional function, we haveThe following theorem has been proved in reference by Li in [18].
Theorem 1. Fordefined in (25) and, we have
Corollary 1. Fordefined in (25),This corollary can be obtained similarly as Theorem 1, where we omit it.
Let be the solution of (1) and be the numerical solution; then, we haveandAccording to the above lemma, the following theorem can be proved.
Theorem 2. Letand; we have
Proof. Aswe haveAs, for , we haveBy the corollary, we obtainSimilarly, for and , we haveandCombining the identity equations (31), (34), (36), and (37), the conclusion of theorem is obtained.
4. Numerical Examples
Example 1. The SDEandunder condition the analysis solution isTables 1 and 2 show the errors of the LBRCM for equidistant nodes of space variables and time variables.
Tables 3 and 4 show the errors of the LBRCM for quasi-equidistant nodes of space variables and time variables.
Example 2. The SDEandunder condition ; the analysis solution isIn Figures 1 and 2, the error estimate of equidistant and quasi-equidistant nodes with is presented. It can be seen from Figure 2 that the barycentric rational interpolation collocation method has higher accuracy in both quasi-equidistant and equidistant nodes conditions.
Tables 5 and 6 show the errors of the LBRCM for equidistant nodes of space variables and time variables.
Tables 7 and 8 show the errors of the LBRCM for quasi-equidistant nodes of space variables and time variables.
5. Conclusion
In this paper, the LBRCM have been constructed to solve SDE, while the time variable and space variable are obtained at the same time. Numerical solution confirms the theorem analysis.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
This manuscript was written by Peicheng Zhao and Yongling Cheng. Some checks of grammar were given by Yongling Cheng.
Acknowledgments
The work of Yongling Cheng was supported by Natural Science Foundation of Hebei Province (Grant no. A2019209533).