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A Parametric Method Optimised for the Solution of the (2+1)-Dimensional Nonlinear Schrödinger Equation

Author

Listed:
  • Zacharias A. Anastassi

    (Institute of Artificial Intelligence, School of Computer Science and Informatics, De Montfort University, Leicester LE1 9BH, UK)

  • Athinoula A. Kosti

    (Institute of Artificial Intelligence, School of Computer Science and Informatics, De Montfort University, Leicester LE1 9BH, UK)

  • Mufutau Ajani Rufai

    (Department of Mathematics, University of Bari Aldo Moro, 70125 Bari, Italy)

Abstract

We investigate the numerical solution of the nonlinear Schrödinger equation in two spatial dimensions and one temporal dimension. We develop a parametric Runge–Kutta method with four of their coefficients considered as free parameters, and we provide the full process of constructing the method and the explicit formulas of all other coefficients. Consequently, we produce an adaptable method with four degrees of freedom, which permit further optimisation. In fact, with this methodology, we produce a family of methods, each of which can be tailored to a specific problem. We then optimise the new parametric method to obtain an optimal Runge–Kutta method that performs efficiently for the nonlinear Schrödinger equation. We perform a stability analysis, and utilise an exact dark soliton solution to measure the global error and mass error of the new method with and without the use of finite difference schemes for the spatial semi-discretisation. We also compare the efficiency of the new method and other numerical integrators, in terms of accuracy versus computational cost, revealing the superiority of the new method. The proposed methodology is general and can be applied to a variety of problems, without being limited to linear problems or problems with oscillatory/periodic solutions.

Suggested Citation

  • Zacharias A. Anastassi & Athinoula A. Kosti & Mufutau Ajani Rufai, 2023. "A Parametric Method Optimised for the Solution of the (2+1)-Dimensional Nonlinear Schrödinger Equation," Mathematics, MDPI, vol. 11(3), pages 1-17, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:609-:d:1046897
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    References listed on IDEAS

    as
    1. Yonglei Fang & Yanping Yang & Xiong You & Lei Ma, 2020. "Modified THDRK methods for the numerical integration of the Schrödinger equation," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 31(10), pages 1-11, October.
    2. Vladislav N. Kovalnogov & Ruslan V. Fedorov & Yuri A. Khakhalev & Theodore E. Simos & Charalampos Tsitouras, 2021. "A Neural Network Technique for the Derivation of Runge–Kutta Pairs Adjusted for Scalar Autonomous Problems," Mathematics, MDPI, vol. 9(16), pages 1-10, August.
    3. N. A. Ahmad & N. Senu & F. Ismail, 2017. "Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs," Mathematical Problems in Engineering, Hindawi, vol. 2017, pages 1-11, February.
    4. Houssem Jerbi & Sondess Ben Aoun & Mohamed Omri & Theodore E. Simos & Charalampos Tsitouras, 2022. "A Neural Network Type Approach for Constructing Runge–Kutta Pairs of Orders Six and Five That Perform Best on Problems with Oscillatory Solutions," Mathematics, MDPI, vol. 10(5), pages 1-10, March.
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