Startseite A Class of Meshless Structure-Preserving Algorithms for the Nonlinear Schrödinger Equation
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

A Class of Meshless Structure-Preserving Algorithms for the Nonlinear Schrödinger Equation

  • Jialing Wang EMAIL logo , Zhengting Zhou und Zhoujin Lin
Veröffentlicht/Copyright: 14. November 2024

Abstract

This paper aims to give a unified construction framework of meshless structure-preserving algorithms to solve the d-dimensional ( d = 1 or 2) nonlinear Schrödinger equation. Based on the method of lines, we first derive a finite-dimensional Hamiltonian system by using the radial basis function method of the quasi-interpolation and the technique of left-multiplying a diagonal matrix to discretize the space direction. Then suitable geometric numerical integrations can be used to discretize the time direction, which yields a class of meshless structure-preserving algorithms. In addition to the construction, the structure-preserving properties and their proofs are also provided in detail. Besides the uniform and nonuniform grids, the numerical experiments on the random grids are also emphasized to verify the theoretical research well, which is of great significance for scattering points based on the characteristics of actual problems.

Award Identifier / Grant number: 11801277

Funding statement: Jialing Wang was supported by the National Natural Science Foundation of China (under Grant No. 11801277).

References

[1] S. Abbasbandy, H. Roohani Ghehsareh and I. Hashim, A meshfree method for the solution of two-dimensional cubic nonlinear Schrödinger equation, Eng. Anal. Bound. Elem. 37 (2013), no. 6, 885–898. 10.1016/j.enganabound.2013.03.006Suche in Google Scholar

[2] W. Bao and Y. Cai, Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator, SIAM J. Numer. Anal. 50 (2012), no. 2, 492–521. 10.1137/110830800Suche in Google Scholar

[3] W. Bao, S. Jin and P. A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys. 175 (2002), no. 2, 487–524. 10.1006/jcph.2001.6956Suche in Google Scholar

[4] Q. Chang, E. Jia and W. Sun, Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys. 148 (1999), no. 2, 397–415. 10.1006/jcph.1998.6120Suche in Google Scholar

[5] M. Dehghan and D. Mirzaei, Numerical solution to the unsteady two-dimensional Schrödinger equation using meshless local boundary integral equation method, Internat. J. Numer. Methods Engrg. 76 (2008), no. 4, 501–520. 10.1002/nme.2338Suche in Google Scholar

[6] M. Dehghan and A. Shokri, A meshless method for numerical solution of the one-dimensional wave equation with an integral condition using radial basis functions, Numer. Algorithms 52 (2009), no. 3, 461–477. 10.1007/s11075-009-9293-0Suche in Google Scholar

[7] Y. Duan and F. Rong, A numerical scheme for nonlinear Schrödinger equation by MQ quasi-interpolation, Eng. Anal. Bound. Elem. 37 (2013), no. 1, 89–94. 10.1016/j.enganabound.2012.08.006Suche in Google Scholar

[8] K. Feng and M. Z. Qin, The symplectic methods for the computation of Hamiltonian equations, Numerical Methods for Partial Differential Equations, Lecture Notes in Math. 1297, Springer, Berlin (1987), 1–37. 10.1007/BFb0078537Suche in Google Scholar

[9] W. Gao and Z. Wu, Approximation orders and shape preserving properties of the multiquadric trigonometric B-spline quasi-interpolant, Comput. Math. Appl. 69 (2015), no. 7, 696–707. 10.1016/j.camwa.2015.02.008Suche in Google Scholar

[10] Y. Gong, Q. Wang, Y. Wang and J. Cai, A conservative Fourier pseudo-spectral method for the nonlinear Schrödinger equation, J. Comput. Phys. 328 (2017), 354–370. 10.1016/j.jcp.2016.10.022Suche in Google Scholar

[11] O. Gonzalez, Time integration and discrete Hamiltonian systems, J. Nonlinear Sci. 6 (1996), no. 5, 449–467. 10.1007/BF02440162Suche in Google Scholar

[12] A. Habibirad, E. Hesameddini and A. Taleei, An efficient meshless method for solving multi-dimensional nonlinear Schrödinger equation, Iran. J. Sci. Technol. Trans. A Sci. 44 (2020), no. 3, 749–761. 10.1007/s40995-020-00864-wSuche in Google Scholar

[13] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, 2nd ed., Springer Ser. Comput. Math. 31, Springer, Berlin, 2006. Suche in Google Scholar

[14] O. Karakashian, G. D. Akrivis and V. A. Dougalis, On optimal order error estimates for the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 30 (1993), no. 2, 377–400. 10.1137/0730018Suche in Google Scholar

[15] L. Kong, Y. Wu, Z. Liu and P. Wang, Pointwise second order convergence of structure-preserving scheme for the triple-coupled nonlinear Schrödinger equations, Comput. Math. Appl. 154 (2024), 91–102. 10.1016/j.camwa.2023.11.002Suche in Google Scholar

[16] E. Larsson and B. Fornberg, A new class of energy-preserving numerical integration methods, J. Phys. 41 (2008), 1–7. Suche in Google Scholar

[17] C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp. 77 (2008), no. 264, 2141–2153. 10.1090/S0025-5718-08-02101-7Suche in Google Scholar

[18] L. Ma and Z. Wu, Approximation to the k-th derivatives by multiquadric quasi-interpolation method, J. Comput. Appl. Math. 231 (2009), no. 2, 925–932. 10.1016/j.cam.2009.05.017Suche in Google Scholar

[19] M. Mongillo, Choosing basis functions and shape parameters for radial basis function methods, SIAM Undergrad. Res. 4 (2011), no. 190–209, 2–6. 10.1137/11S010840Suche in Google Scholar

[20] G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A 41 (2008), no. 4, Article ID 045206. 10.1088/1751-8113/41/4/045206Suche in Google Scholar

[21] S. Rippa, An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv. Comput. Math. 11 (1999), 193–210. 10.1023/A:1018975909870Suche in Google Scholar

[22] C. Stuart, R. Enns, S. Rangnekar and S. Sukhpal, Quasi-soliton and other behaviour of the nonlinear cubic-quintic Schrödinger equation, Canad. J. Phys. 64 (1986), no. 311, 311–315. 10.1139/p86-054Suche in Google Scholar

[23] C. Sun and J. Wang, Several difference schemes for nonlinear Schrödinger equation, J. Huaqiao Univ. 42 (2021), no. 4, 551–560. Suche in Google Scholar

[24] Z. Sun, A meshless symplectic method for two-dimensional nonlinear Schrödinger equations based on radial basis function approximation, Eng. Anal. Bound. Elem. 104 (2019), 1–7. 10.1016/j.enganabound.2019.03.014Suche in Google Scholar

[25] Z. Sun, A conservative scheme for two-dimensional Schrödinger equation based on multiquadric trigonometric quasi-interpolation approach, Appl. Math. Comput. 423 (2022), Article ID 126996. 10.1016/j.amc.2022.126996Suche in Google Scholar

[26] Z. Sun and W. Gao, A meshless scheme for Hamiltonian partial differential equations with conservation properties, Appl. Numer. Math. 119 (2017), 115–125. 10.1016/j.apnum.2017.04.005Suche in Google Scholar

[27] I. Talanov, Self focusing of wave beams in nonlinear media, Jetp Lett. 2 (1965), no. 55, 138–141. Suche in Google Scholar

[28] J. Wang, A new error analysis of Crank–Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation, J. Sci. Comput. 60 (2014), no. 2, 390–407. 10.1007/s10915-013-9799-4Suche in Google Scholar

[29] T. Wang, J. Wang and B. Guo, Two completely explicit and unconditionally convergent Fourier pseudo-spectral methods for solving the nonlinear Schrödinger equation, J. Comput. Phys. 404 (2020), Article ID 109116. 10.1016/j.jcp.2019.109116Suche in Google Scholar

[30] T. Wang and X. Zhao, Unconditional L -convergence of two compact conservative finite difference schemes for the nonlinear Schrödinger equation in multi-dimensions, Calcolo 55 (2018), no. 3, 34–60. 10.1007/s10092-018-0277-0Suche in Google Scholar

[31] Y. Wang, B. Wang and M. Qin, Local structure-preserving algorithms for partial differential equations, Sci. China Ser. A 51 (2008), no. 11, 2115–2136. 10.1007/s11425-008-0046-7Suche in Google Scholar

[32] G. Whitham and G. Fowler, Linear and nonlinear waves, Amer. Inst. Phys. 6 (1975), no. 55, 55–56. 10.1063/1.3069011Suche in Google Scholar

[33] Z. Wu and S. Zhang, Conservative multiquadric quasi-interpolation method for Hamiltonian wave equations, Eng. Anal. Bound. Elem. 37 (2013), no. 7–8, 1052–1058. 10.1016/j.enganabound.2013.04.011Suche in Google Scholar

[34] Z. M. Wu, Radial basis functions in scattered data interpolation and the meshless method of numerical solution of PDEs, Gongcheng Shuxue Xuebao 19 (2002), no. 2, 1–12. Suche in Google Scholar

[35] Z. M. Wu and J. P. Liu, Generalized Strang-Fix condition for scattered data quasi-interpolation, Adv. Comput. Math. 23 (2005), no. 1–2, 201–214. 10.1007/s10444-004-1832-6Suche in Google Scholar

[36] Z. M. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993), no. 1, 13–27. 10.1093/imanum/13.1.13Suche in Google Scholar

[37] Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. Comput. Phys. 205 (2005), no. 1, 72–97. 10.1016/j.jcp.2004.11.001Suche in Google Scholar

[38] Y. Zhang, W. Bao and H. Li, Dynamics of rotating two-component Bose–Einstein condensates and its efficient computation, Phys. D 234 (2007), no. 1, 49–69. 10.1016/j.physd.2007.06.026Suche in Google Scholar

[39] G. Zhong and J. E. Marsden, Lie–Poisson Hamilton–Jacobi theory and Lie–Poisson integrators, Phys. Lett. A 133 (1988), no. 3, 134–139. 10.1016/0375-9601(88)90773-6Suche in Google Scholar

Received: 2023-09-28
Revised: 2024-09-01
Accepted: 2024-09-23
Published Online: 2024-11-14

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 27.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/cmam-2023-0213/html
Button zum nach oben scrollen