IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v423y2022ics0096300322000820.html
   My bibliography  Save this article

A conservative scheme for two-dimensional Schrödinger equation based on multiquadric trigonometric quasi-interpolation approach

Author

Listed:
  • Sun, Zhengjie

Abstract

In this paper, we propose a conservative scheme to solve the time-dependent two-dimensional nonlinear Schrödinger equation, which plays an important role in many fields of physics. We discretize the equation using the multiquadric trigonometric quasi-interpolation method in space and the Crank–Nicolson scheme in time. The quasi-interpolant is constructed through a linear combination of function values and node translations of a kernel function. It is very simple to compute since it doesn’t need to solve any linear system. We adopt two different approaches to approximate second order spatial derivatives and analyze the symmetric or anti-symmetric properties of differentiation matrices. Moreover, the convergence and conservation properties including the total mass and energy conservation laws are also investigated in detail. Numerical experiments are performed to illustrate the accuracy and efficiency of the proposed method.

Suggested Citation

  • 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).
  • Handle: RePEc:eee:apmaco:v:423:y:2022:i:c:s0096300322000820
    DOI: 10.1016/j.amc.2022.126996
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300322000820
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.amc.2022.126996?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. Frasca-Caccia, Gianluca & Hydon, Peter E., 2021. "Numerical preservation of multiple local conservation laws," Applied Mathematics and Computation, Elsevier, vol. 403(C).
    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.
    3. Dehghan, Mehdi & Shokri, Ali, 2008. "A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 700-715.
    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. Nikolay A. Kudryashov, 2023. "Hamiltonians of the Generalized Nonlinear Schrödinger Equations," Mathematics, MDPI, vol. 11(10), pages 1-12, May.

    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. 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.
    2. 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.
    3. Saffarian, Marziyeh & Mohebbi, Akbar, 2021. "Numerical solution of two and three dimensional time fractional damped nonlinear Klein–Gordon equation using ADI spectral element method," Applied Mathematics and Computation, Elsevier, vol. 405(C).
    4. Saberi Zafarghandi, Fahimeh & Mohammadi, Maryam & Babolian, Esmail & Javadi, Shahnam, 2019. "Radial basis functions method for solving the fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 342(C), pages 224-246.
    5. 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.
    6. Gupta, A.K. & Saha Ray, S., 2017. "On the solitary wave solution of fractional Kudryashov–Sinelshchikov equation describing nonlinear wave processes in a liquid containing gas bubbles," Applied Mathematics and Computation, Elsevier, vol. 298(C), pages 1-12.
    7. Frasca-Caccia, Gianluca & Hydon, Peter E., 2021. "Numerical preservation of multiple local conservation laws," Applied Mathematics and Computation, Elsevier, vol. 403(C).
    8. Adak, D. & Natarajan, S., 2020. "Virtual element method for semilinear sine–Gordon equation over polygonal mesh using product approximation technique," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 224-243.
    9. Deng, Dingwen & Wu, Qiang, 2023. "Accuracy improvement of a Predictor–Corrector compact difference scheme for the system of two-dimensional coupled nonlinear wave equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 223-249.
    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. Li, Meng & Fei, Mingfa & Wang, Nan & Huang, Chengming, 2020. "A dissipation-preserving finite element method for nonlinear fractional wave equations on irregular convex domains," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 177(C), pages 404-419.
    13. 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.
    14. 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).
    15. 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.
    16. 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.
    17. 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.
    18. Jiong Weng & Xiaojing Liu & Youhe Zhou & Jizeng Wang, 2021. "A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations," Mathematics, MDPI, vol. 9(22), pages 1-17, November.
    19. Luigi Brugnano & Gianluca Frasca-Caccia & Felice Iavernaro, 2019. "Line Integral Solution of Hamiltonian PDEs," Mathematics, MDPI, vol. 7(3), pages 1-28, March.
    20. 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).

    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:423:y:2022:i:c:s0096300322000820. 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.