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

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  • 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.

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  • 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
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    References listed on IDEAS

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    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.
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    Cited by:

    1. Nikolay A. Kudryashov, 2023. "Hamiltonians of the Generalized Nonlinear Schrödinger Equations," Mathematics, MDPI, vol. 11(10), pages 1-12, May.

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