IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v318y2018icp3-18.html
   My bibliography  Save this article

Energy-conserving methods for the nonlinear Schrödinger equation

Author

Listed:
  • Barletti, L.
  • Brugnano, L.
  • Frasca Caccia, G.
  • Iavernaro, F.

Abstract

In this paper, we further develop recent results in the numerical solution of Hamiltonian partial differential equations (PDEs) (Brugnano et al., 2015), by means of energy-conserving methods in the class of Line Integral Methods, in particular, the Runge–Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). We shall use HBVMs for solving the nonlinear Schrödinger equation (NLSE), of interest in many applications. We show that the use of energy-conserving methods, able to conserve a discrete counterpart of the Hamiltonian functional, confers more robustness on the numerical solution of such a problem.

Suggested Citation

  • Barletti, L. & Brugnano, L. & Frasca Caccia, G. & Iavernaro, F., 2018. "Energy-conserving methods for the nonlinear Schrödinger equation," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 3-18.
    • Handle: RePEc:eee:apmaco:v:318:y:2018:i:c:p:3-18
    DOI: 10.1016/j.amc.2017.04.018
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300317302709
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2017.04.018?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Islas, A.L. & Schober, C.M., 2005. "Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 69(3), pages 290-303.
    2. Brugnano, L. & Frasca Caccia, G. & Iavernaro, F., 2015. "Energy conservation issues in the numerical solution of the semilinear wave equation," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 842-870.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Frasca-Caccia, Gianluca & Hydon, Peter E., 2021. "Numerical preservation of multiple local conservation laws," Applied Mathematics and Computation, Elsevier, vol. 403(C).
    2. Vyacheslav Trofimov & Maria Loginova & Mikhail Fedotov & Daniil Tikhvinskii & Yongqiang Yang & Boyuan Zheng, 2022. "Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation," Mathematics, MDPI, vol. 10(11), pages 1-24, June.
    3. Li, Haochen & Jiang, Chaolong & Lv, Zhongquan, 2018. "A Galerkin energy-preserving method for two dimensional nonlinear Schrödinger equation," Applied Mathematics and Computation, Elsevier, vol. 324(C), pages 16-27.
    4. Auzinger, Winfried & Hofstätter, Harald & Koch, Othmar & Kropielnicka, Karolina & Singh, Pranav, 2019. "Time adaptive Zassenhaus splittings for the Schrödinger equation in the semiclassical regime," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.
    5. Amodio, Pierluigi & Brugnano, Luigi & Iavernaro, Felice, 2019. "A note on the continuous-stage Runge–Kutta(–Nyström) formulation of Hamiltonian Boundary Value Methods (HBVMs)," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
    6. Vyacheslav Trofimov & Maria Loginova, 2021. "Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants," Mathematics, MDPI, vol. 9(21), pages 1-26, October.
    7. Luigi Brugnano & Gianluca Frasca-Caccia & Felice Iavernaro, 2019. "Line Integral Solution of Hamiltonian PDEs," Mathematics, MDPI, vol. 7(3), pages 1-28, March.
    8. Huang, Yifei & Peng, Gang & Zhang, Gengen & Zhang, Hong, 2023. "High-order Runge–Kutta structure-preserving methods for the coupled nonlinear Schrödinger–KdV equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 603-618.
    9. Zhang, Fan & Sun, Hai-Wei & Sun, Tao, 2024. "Efficient and unconditionally energy stable exponential-SAV schemes for the phase field crystal equation," Applied Mathematics and Computation, Elsevier, vol. 470(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Frasca-Caccia, Gianluca & Hydon, Peter E., 2021. "Numerical preservation of multiple local conservation laws," Applied Mathematics and Computation, Elsevier, vol. 403(C).
    2. Trofimov, Vyacheslav A. & Stepanenko, Svetlana & Razgulin, Alexander, 2021. "Conservation laws of femtosecond pulse propagation described by generalized nonlinear Schrödinger equation with cubic nonlinearity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 366-396.
    3. Liu, Changying & Wu, Xinyuan & Shi, Wei, 2018. "New energy-preserving algorithms for nonlinear Hamiltonian wave equation equipped with Neumann boundary conditions," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 588-606.
    4. Almushaira, Mustafa, 2023. "Efficient energy-preserving eighth-order compact finite difference schemes for the sine-Gordon equation," Applied Mathematics and Computation, Elsevier, vol. 451(C).
    5. Yang, Yanhong & Wang, Yushun & Song, Yongzhong, 2018. "A new local energy-preserving algorithm for the BBM equation," Applied Mathematics and Computation, Elsevier, vol. 324(C), pages 119-130.
    6. Xie, Jianqiang & Zhang, Zhiyue, 2019. "An analysis of implicit conservative difference solver for fractional Klein–Gordon–Zakharov system," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 153-166.
    7. Sun, Zhengjie, 2022. "A conservative scheme for two-dimensional Schrödinger equation based on multiquadric trigonometric quasi-interpolation approach," Applied Mathematics and Computation, Elsevier, vol. 423(C).
    8. Luigi Brugnano & Gianluca Frasca-Caccia & Felice Iavernaro, 2019. "Line Integral Solution of Hamiltonian PDEs," Mathematics, MDPI, vol. 7(3), pages 1-28, March.
    9. Jiang, Chaolong & Sun, Jianqiang & Li, Haochen & Wang, Yifan, 2017. "A fourth-order AVF method for the numerical integration of sine-Gordon equation," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 144-158.
    10. Zhang, Gengen, 2021. "Two conservative and linearly-implicit compact difference schemes for the nonlinear fourth-order wave equation," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    11. Amodio, Pierluigi & Brugnano, Luigi & Iavernaro, Felice, 2019. "A note on the continuous-stage Runge–Kutta(–Nyström) formulation of Hamiltonian Boundary Value Methods (HBVMs)," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
    12. Chen, Hao & Yang, Yeru, 2021. "Generalized Störmer-Cowell methods with efficient iterative solver for large-scale second-order stiff semilinear systems," Applied Mathematics and Computation, Elsevier, vol. 400(C).
    13. Brugnano, L. & Frasca Caccia, G. & Iavernaro, F., 2015. "Energy conservation issues in the numerical solution of the semilinear wave equation," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 842-870.
    14. Aydın, Ayhan, 2009. "Multisymplectic integration of N-coupled nonlinear Schrödinger equation with destabilized periodic wave solutions," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 735-751.
    15. Martin-Vergara, Francisca & Rus, Francisco & Villatoro, Francisco R., 2019. "Padé numerical schemes for the sine-Gordon equation," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 232-243.
    16. Achouri, Talha, 2019. "Conservative finite difference scheme for the nonlinear fourth-order wave equation," Applied Mathematics and Computation, Elsevier, vol. 359(C), pages 121-131.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:318:y:2018:i:c:p:3-18. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.