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A conservative scheme with optimal error estimates for a multidimensional space-fractional Gross–Pitaevskii equation

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Języki publikacji
EN
Abstrakty
EN
The present work departs from an extended form of the classical multi-dimensional Gross–Pitaevskii equation, which considers fractional derivatives of the Riesz type in space, a generalized potential function and angular momentum rotation. It is well known that the classical system possesses functionals which are preserved throughout time. It is easy to check that the generalized fractional model considered in this work also possesses conserved quantities, whence the development of conservative and efficient numerical schemes is pragmatically justified. Motivated by these facts, we propose a finite-difference method based on weighted-shifted Grünwald differences to approximate the solutions of the generalized Gross–Pitaevskii system. We provide here a discrete extension of the uniform Sobolev inequality to multiple dimensions, and show that the proposed method is capable of preserving discrete forms of the mass and the energy of the model. Moreover, we establish thoroughly the stability and the convergence of the technique, and provide some illustrative simulations to show that the method is capable of preserving the total mass and the total energy of the generalized system.
Rocznik
Strony
713--723
Opis fizyczny
Bibliogr. 38 poz., rys.
Twórcy
  • Department of Computational Mathematics and Computer Science, Ural Federal University, 19 Mira St., Yekaterinburg 620002, Russia; Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt, ahmed.hendy@fsc.bu.edu.eg
  • Department of Mathematics and Physics, Autonomous University of Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes 20131, Mexico, jemacias@correo.uaa.mx
Bibliografia
  • [1] Alikhanov, A.A. (2015). A new difference scheme for the time fractional diffusion equation, Journal of Computational Physics 280: 424–438.
  • [2] Alves, C.O. and Miyagaki, O.H. (2016). Existence and concentration of solution for a class of fractional elliptic equation in Rn via penalization method, Calculus of Variations and Partial Differential Equations 55(3): 47.
  • [3] Antoine, X., Tang, Q. and Zhang, Y. (2016). On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross–Pitaevskii equations with rotation term and nonlocal nonlinear interactions, Journal of Computational Physics 325: 74–97.
  • [4] Bao, W. and Cai, Y. (2013). Optimal error estimates of finite difference methods for the Gross–Pitaevskii equation with angular momentum rotation, Mathematics of Computation 82(281): 99–128.
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  • [16] Macías-Díaz, J.E. (2017). A structure-preserving method for a class of nonlinear dissipative wave equations with Riesz space-fractional derivatives, Journal of Computational Physics 351: 40–58.
  • [17] Macías-Díaz, J.E. (2018). An explicit dissipation-preserving method for Riesz space-fractional nonlinear wave equations in multiple dimensions, Communications in Nonlinear Science and Numerical Simulation 59: 67–87.
  • [18] Macías-Díaz, J.E. (2019). On the solution of a Riesz space-fractional nonlinear wave equation through an efficient and energy-invariant scheme, International Journal of Computer Mathematics 96(2): 337–361.
  • [19] Matsuo, T. and Furihata, D. (2001). Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations, Journal of Computational Physics 171(2): 425–447.
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  • [22] Pimenov, V.G. and Hendy, A.S. (2017). A numerical solution for a class of time fractional diffusion equations with delay, International Journal of Applied Mathematics and Computer Science 27(3): 477–488, DOI: 10.1515/amcs-2017-0033.
  • [23] Pimenov, V.G., Hendy, A.S. and De Staelen, R.H. (2017). On a class of non-linear delay distributed order fractional diffusion equations, Journal of Computational and Applied Mathematics 318: 433–443.
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  • [34] Tian, W., Zhou, H. and Deng, W. (2015). A class of second order difference approximations for solving space fractional diffusion equations, Mathematics of Computation 84(294): 1703–1727.
  • [35] Wang, P., Huang, C. and Zhao, L. (2016). Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation, Journal of Computational and Applied Mathematics 306: 231–247.
  • [36] Wang, T., Jiang, J. and Xue, X. (2018). Unconditional and optimal H1 error estimate of a Crank–Nicolson finite difference scheme for the Gross–Pitaevskii equation with an angular momentum rotation term, Journal of Mathematical Analysis and Applications 459(2): 945–958.
  • [37] Wang, T. and Zhao, X. (2014). Optimal l∞ error estimates of finite difference methods for the coupled Gross–Pitaevskii equations in high dimensions, Science China Mathematics 57(10): 2189–2214.
  • [38] Ye, H., Liu, F. and Anh, V. (2015). Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains, Journal of Computational Physics 298: 652–660.
Uwagi
PL
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-12c0528e-60ea-4bf2-a682-ba1c8a22568a
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