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

Hendy, Ahmed S. ; Macías-Díaz, Jorge E.

International Journal of Applied Mathematics and Computer Science, Tome 29 (2019) no. 4, pp. 713-723.

Résumé

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.

Détail

Keywords: generalized Gross–Pitaevskii system, Riesz fractional diffusion, Sobolev inequality, conservative method, optimal error bounds
Mots-clés : równanie Grossa-Pitaevskiego, nierówność Sobolewa, metoda konserwatywna, optymalna granica błędu

Bibliographie

[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.

[5] Ben-Yu, G., Pascual, P.J., Rodriguez, M.J. and Vázquez, L. (1986). Numerical solution of the sine-Gordon equation, Applied Mathematics and Computation 18(1): 1–14.

[6] Bhrawy, A.H. and Abdelkawy, M.A. (2015). A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations, Journal of Computational Physics 294: 462–483.

[7] El-Ajou, A., Arqub, O.A. and Momani, S. (2015). Approximate analytical solution of the nonlinear fractional KdV–Burgers equation: A new iterative algorithm, Journal of Computational Physics 293: 81–95.

[8] Fei, Z. and Vázquez, L. (1991). Two energy conserving numerical schemes for the sine-Gordon equation, Applied Mathematics and Computation 45(1): 17–30.

[9] Furihata, D. (2001). Finite-difference schemes for nonlinear wave equation that inherit energy conservation property, Journal of Computational and Applied Mathematics 134(1): 37–57.

[10] Glöckle, W.G. and Nonnenmacher, T.F. (1995). A fractional calculus approach to self-similar protein dynamics, Biophysical Journal 68(1): 46–53.

[11] Gross, E.P. (1961). Structure of a quantized vortex in boson systems, Il Nuovo Cimento (1955-1965) 20(3): 454–477.

[12] Iannizzotto, A., Liu, S., Perera, K. and Squassina, M. (2016). Existence results for fractional p-Laplacian problems via Morse theory, Advances in Calculus of Variations 9(2): 101–125.

[13] Kaczorek, T. (2015). Positivity and linearization of a class of nonlinear continuous-time systems by state feedbacks, International Journal of Applied Mathematics and Computer Science 25(4): 827–831, DOI: 10.1515/amcs-2015-0059.

[14] Koeller, R. (1984). Applications of fractional calculus to the theory of viscoelasticity, ASME Transactions: Journal of Applied Mechanics 51: 299–307.

[15] Liu, F., Zhuang, P., Turner, I., Anh, V. and Burrage, K. (2015). A semi-alternating direction method for a 2-D fractional FitzHugh–Nagumo monodomain model on an approximate irregular domain, Journal of Computational Physics 293: 252–263.

[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.

[20] Namias, V. (1980). The fractional order Fourier transform and its application to quantum mechanics, IMA Journal of Applied Mathematics 25(3): 241–265.

[21] Oprzędkiewicz, K., Gawin, E. and Mitkowski, W. (2016). Modeling heat distribution with the use of a non-integer order, state space model, International Journal of Applied Mathematics and Computer Science 26(4): 749–756, DOI: 10.1515/amcs-2016-0052.

[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.

[24] Pitaevskii, L. (1961). Vortex lines in an imperfect Bose gas, Soviet Physics JETP 13(2): 451–454.

[25] Povstenko, Y. (2009). Theory of thermoelasticity based on the space-time-fractional heat conduction equation, Physica Scripta 2009(T136): 014017.

[26] Rakkiyappan, R., Cao, J. and Velmurugan, G. (2015). Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays, IEEE Transactions on Neural Networks and Learning Systems 26(1): 84–97.

[27] Raman, C., Köhl, M., Onofrio, R., Durfee, D., Kuklewicz, C., Hadzibabic, Z. and Ketterle, W. (1999). Evidence for a critical velocity in a Bose–Einstein condensed gas, Physical Review Letters 83(13): 2502.

[28] Scalas, E., Gorenflo, R. and Mainardi, F. (2000). Fractional calculus and continuous-time finance, Physica A: Statistical Mechanics and Its Applications 284(1): 376–384.

[29] Strauss, W. and Vazquez, L. (1978). Numerical solution of a nonlinear Klein–Gordon equation, Journal of Computational Physics 28(2): 271–278.

[30] Su, N., Nelson, P.N. and Connor, S. (2015). The distributed-order fractional diffusion-wave equation of groundwater flow: Theory and application to pumping and slug tests, Journal of Hydrology 529(3): 1262–1273.

[31] Tang, Y.-F., Vázquez, L., Zhang, F. and Pérez-García, V. (1996). Symplectic methods for the nonlinear Schrödinger equation, Computers Mathematics with Applications 32(5): 73–83.

[32] Tarasov, V.E. (2006). Continuous limit of discrete systems with long-range interaction, Journal of Physics A: Mathematical and General 39(48): 14895.

[33] Tarasov, V.E. and Zaslavsky, G.M. (2008). Conservation laws and HamiltonâĂŹs equations for systems with long-range interaction and memory, Communications in Nonlinear Science and Numerical Simulation 13(9): 1860–1878.

[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.