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Exponential time differencing schemes for the 3-coupled nonlinear fractional Schrödinger equation
- Xiao Liang^{1}Email authorView ORCID ID profile and
- Harish Bhatt^{2}
https://doi.org/10.1186/s13662-018-1936-9
© The Author(s) 2018
- Received: 20 August 2018
- Accepted: 16 December 2018
- Published: 29 December 2018
Abstract
Two modified exponential time differencing schemes based on the Fourier spectral method are developed to solve the 3-coupled nonlinear fractional Schrödinger equation. We compare the stability of the schemes by plotting their stability regions. The local truncation errors of the time integrators are proved to be fourth-order. Numerical experiments illustrating the solution to the equations with various parameters and the mass conservation results of the numerical methods are carried out.
Keywords
- Space-fractional
- Nonlinear Schrödinger equations
- Exponential time differencing
- Spectral method
- Fractional Laplacian
MSC
- 35Q55
- 65M70
- 65M12
1 Introduction
During the last few years, the nonlinear fractional Schrödinger equation (NFSE) has been widely used in modeling physical phenomena such as the propagation of waves in optics and hydrodynamics, see [1–4] for details. Various numerical methods have been developed and used for solving the fractional Schrödinger equations. For example, a series of difference schemes have been proposed by Wang and Huang [5–7] for the one- and two-dimensional space-fractional nonlinear Schrödinger equations. A Fourier spectral method has been developed by Duo and Zhang [8] considering the fractional parabolic equations based on periodic or Neumann boundary conditions. Various numerical methods have been proposed by Dehghan et al. [9–14] for solving the systems of one- and multi-dimensional nonlinear Schrödinger equations efficiently.
Although the 2-coupled NFSE was considered in a number of research articles, the 3CNFSE was rarely mentioned. In fact, the 3CNFSE is important in modeling the propagation of periodic solitary waves with perturbation in space, and this kind of model cannot be replaced by the 2-coupled NFSE. This inspired us to find some efficient numerical methods for the 3CNFSE.
The outline of this paper is organized as follows. In Sect. 2, we describe the space discretization for the 3CNFSE, which is the Fourier spectral method. In Sect. 3, we propose the description of the modified exponential time differencing schemes as time integrators for the 3CNFSE. In Sect. 4, the stability issues and rates of truncation errors to the proposed methods are analyzed and proved. In Sect. 5, numerical experiments are demonstrated and the mass conservative property of the proposed schemes is indicated. Finally, we give the conclusion in Sect. 6.
2 The Fourier spectral method
3 The exponential time differential procedure
4 Linear analysis of the ETDRK4-P schemes
4.1 Truncation error
4.2 Stability analysis
4.2.1 Amplification symbol
As defined by Yousuf et al. [21], a rational approximation \(R_{r,s}(z)\) to the exponential \(e^{-z}\) is called A-acceptable when \(|R_{r,s}(-z)|<1\) holds for all −z with negative real part. The approximation is called L-acceptable when it is A-acceptable and it also satisfies \(|R_{r,s}(-z)|\rightarrow0\) as \(\mathfrak{R}(-z)\rightarrow-\infty\).
4.3 Stability regions
5 Numerical experiments
5.1 The split-step scheme
5.2 A one-dimensional 3CNFSE
Convergence rates of the ETDRK4-P22 scheme in time
τ | α = 1.2 | α = 1.6 | α = 1.8 | Avg. CPU (s) | |||
---|---|---|---|---|---|---|---|
e(τ) | p | e(τ) | p | e(τ) | p | ||
1/100 | 1.5285e−2 | 3.982 | 1.6338e−2 | 4.049 | 1.6226e−2 | 4.026 | 2.3439 |
1/200 | 9.6741e−4 | 4.011 | 9.8719e−4 | 4.015 | 9.9607e−4 | 4.008 | 4.2552 |
1/400 | 6.0013e−5 | 3.993 | 6.1051e−5 | 3.993 | 6.1906e−5 | 4.010 | 8.0172 |
1/800 | 3.7697e−6 | – | 3.8349e−6 | – | 3.8427e−6 | – | 15.3620 |
1/1600 | – | – | – | – | – | – | 28.3520 |
Convergence rates of the ETDRK4-P13 scheme in time
τ | α = 1.2 | α = 1.6 | α = 1.8 | Avg. CPU (s) | |||
---|---|---|---|---|---|---|---|
e(τ) | p | e(τ) | p | e(τ) | p | ||
1/100 | 1.2388e−2 | 3.989 | 1.3319e−2 | 4.005 | 1.2256e−2 | 3.996 | 2.4631 |
1/200 | 7.8010e−4 | 4.006 | 8.2984e−4 | 4.011 | 7.6792e−4 | 4.021 | 4.7275 |
1/400 | 4.8544e−5 | 4.012 | 5.1479e−5 | 4.007 | 4.7315e−5 | 4.015 | 8.6160 |
1/800 | 3.0095e−6 | – | 3.2014e−6 | – | 2.9261e−6 | – | 15.2435 |
1/1600 | – | – | – | – | – | – | 29.0225 |
α | t = 25 | t = 50 | t = 75 | t = 100 |
---|---|---|---|---|
1.2 | 5.2037e−9 | 4.8303e−10 | 3.0584e−9 | 4.0778e−10 |
1.4 | 7.6506e−10 | 7.6807e−10 | 4.3926e−9 | 8.6931e−10 |
1.6 | 5.8098e−10 | 6.9978e−10 | 3.7575e−9 | 2.7728e−9 |
1.8 | 9.6843e−10 | 1.0083e−9 | 2.8843e−9 | 2.9347e−9 |
2.0 | 3.8854e−10 | 1.3883e−10 | 1.1255e−10 | 7.6353e−11 |
α | t = 25 | t = 50 | t = 75 | t = 100 |
---|---|---|---|---|
1.2 | 4.4709e−10 | 9.6513e−11 | 5.9675e−10 | 3.3412e−10 |
1.4 | 3.9881e−11 | 1.1905e−11 | 2.3539e−11 | 9.1806e−10 |
1.6 | 6.7730e−10 | 8.7725e−11 | 4.6004e−11 | 9.8222e−10 |
1.8 | 4.8611e−10 | 3.8307e−11 | 7.6264e−11 | 5.2848e−10 |
2.0 | 5.4763e−11 | 8.1800e−11 | 8.3620e−11 | 8.6351e−11 |
Stability of the ERK4 method and the ETDRK4-P13 scheme
t | ERK4 (τ = 0.04) | ETDRK4-P13 (τ = 0.04) |
---|---|---|
\(L_{\infty}\) errors | \(L_{\infty}\) errors | |
10 | inf | 0.00082 |
20 | inf | 0.00173 |
30 | inf | 0.00243 |
40 | inf | 0.00328 |
5.3 A two-dimensional 3CNFSE
5.4 A three-dimensional 3CNFSE
6 Conclusion
In this work, two modified exponential time differencing Runge–Kutta schemes, using the Padé approximation to the matrix exponential functions, have been developed for the 3-coupled space-fractional nonlinear Schrödinger equation. The stability issues are discussed by computing the amplification factors and plotting the stability regions. Local truncation errors are calculated to indicate the accuracy of the proposed schemes. Numerical experiments are undertaken on the equations with various α values. The results indicate that the proposed schemes conserve mass of the waves well.
Declarations
Funding
This work was supported by Hubei Provincial Department of Education (B2018158); the Project of Hubei University of Arts and Science (XK2018033).
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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