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A NEW LINEAR AND CONSERVATIVE FINITE DIFFERENCE SCHEME FOR THE GROSS–PITAEVSKII EQUATION WITH ANGULAR MOMENTUM ROTATION

  • JIN CUI (a1) (a2), WENJUN CAI (a1), CHAOLONG JIANG (a3) and YUSHUN WANG (a1)

Abstract

A new linear and conservative finite difference scheme which preserves discrete mass and energy is developed for the two-dimensional Gross–Pitaevskii equation with angular momentum rotation. In addition to the energy estimate method and mathematical induction, we use the lifting technique as well as some well-known inequalities to establish the optimal $H^{1}$ -error estimate for the proposed scheme with no restrictions on the grid ratio. Unlike the existing numerical solutions which are of second-order accuracy at the most, the convergence rate of the numerical solution is proved to be of order $O(h^{4}+\unicode[STIX]{x1D70F}^{2})$ with time step $\unicode[STIX]{x1D70F}$ and mesh size $h$ . Numerical experiments have been carried out to show the efficiency and accuracy of our new method.

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[1] Antoine, X., Bao, W. and Besse, C., “Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations”, Comput. Phys. Comm. 184 (2013) 26212633; doi:10.1016/j.cpc.2013.07.012.
[2] Antoine, X. and Duboscq, R., “GPELab, a Matlab toolbox to solve Gross–Pitaevskii equations I: Computation of stationary solutions”, Comput. Phys. Comm. 185 (2014) 29692991; doi:10.1016/j.cpc.2014.06.026.
[3] Bao, W. and Cai, Y., “Optimal error estimates of finite difference methods for the Gross–Pitaevskii equation with angular momentum rotation”, Math. Comp. 82 (2013) 99128; doi:10.1090/S0025-5718-2012-02617-2.
[4] Bao, W. and Du, Q., “Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow”, SIAM J. Sci. Comput. 25 (2004) 16741697; doi:10.1137/S1064827503422956.
[5] Browder, F. E., “Existence and uniqueness theorems for solutions of nonlinear boundary value problems”, in: Applications of nonlinear partial differential equations, Volume 17 of Proceedings of Symposia in Applied Mathematics (ed. Finn, R.), (American Mathematical Society, Providence, 1965) 2449; doi:10.1090/psapm/017/0197933.
[6] Castin, Y. and Dum, R., “Bose–Einstein condensates with vortices in rotating traps”, Eur. Phys. J. D 7 (1999) 399412; doi:10.1007/s100530050584.
[7] Chang, Q., Jia, E. and Sun, W., “Difference schemes for solving the generalized nonlinear Schrödinger equation”, J. Comput. Phys. 148 (1999) 397415; doi:10.1006/jcph.1998.6120.
[8] Cloot, A., Herbst, B. M. and Weideman, J. A. C., “A numerical study of the nonlinear Schrödinger equation involving quintic terms”, J. Comput. Phys. 86 (1990) 127146; doi:10.1016/0021-9991(90)90094-H.
[9] Dalfovo, F. and Giorgini, S., “Theory of Bose–Einstein condensation in trapped gases”, Rev. Mod. Phys. 71 (1999) 463512; doi:10.1103/RevModPhys.71.463.
[10] Gong, Y. Z., Wang, Q., Wang, Y. S. and Cai, J. X., “A conservative Fourier pseudospectral method for the nonlinear Schrödinger equation”, J. Comput. Phys. 328 (2017) 354370; doi:10.1016/j.jcp.2016.10.022.
[11] Gray, R., “Toeplitz and circulant matrices”, ISL, Technical Report, Stanford University, Stanford, CA, 2002; https://ee.stanford.edu/∼gray/toeplitz.html.
[12] Guo, B. Y., “The convergence of numerical method for nonlinear Schrödinger equation”, J. Comput. Math. 4 (1986) 121130; http://www.global-sci.org/v1/jcm/volumes/v4n2/pdf/042-121.pdf.
[13] Hao, C. C., Hsiao, L. and Li, H. L., “Global well posedness for the Gross–Pitaevskii equation with an angular momentum rotational term”, Math. Methods Appl. Sci. 31 (2008) 655664; doi:10.1002/mma.931.
[14] Henning, P. and Malqvist, A., “The finite element method for the time-dependent Gross–Pitaevskii equation with angular momentum rotation”, SIAM J. Numer. Anal. 55 (2017) 923952; doi:10.1137/15M1009172.
[15] Lees, M., “Approximate solutions of parabolic equations”, J. Soc. Ind. Appl. Math. 7 (1959) 167183; doi:10.1137/0107015.
[16] Liao, H. L. and Sun, Z. Z., “Error estimate of fourth-order compact scheme for linear Schrödinger equations”, SIAM J. Numer. Anal. 47 (2010) 43814401; doi:10.1137/080714907.
[17] Lieb, E. H. and Seiringer, R., “Derivation of the Gross–Pitaevskii equation for rotating Bose gases”, Commun. Math. Phys. 264 (2006) 505537; doi:10.1007/s00220-006-1524-9.
[18] Madison, K. W., Chevy, F., Wohlleben, W. and Dalibard, J., “Vortex formation in a stirred Bose–Einstein condensate”, Phys. Rev. Lett. 84 (2000) 806809; doi:10.1103/PhysRevLett.84.806.
[19] Matthews, M. R., Anderson, B. P., Haljan, P. C., Hall, D. S., Wieman, C. E. and Cornell, E. A., “Vortices in a Bose–Einstein condensate”, Phys. Rev. Lett. 83 (1999) 24982501; doi:10.1103/PhysRevLett.83.2498.
[20] Pitaevskii, L. and Stringary, S., Bose–Einstein condensation, Volume 116 of International Series of Monographs on Physics (Clarendon Press, Oxford, 2003).
[21] Shen, J., “A new dual-Petrov–Galerkin method for third and higher odd-order differential equations: application to the KDV equation”, SIAM J. Numer. Anal. 41 (2003) 15951619; doi:10.1137/S0036142902410271.
[22] Sun, W. W. and Wang, J. L., “Optimal error analysis of Crank–Nicolson schemes for a coupled nonlinear Schrödinger system in 3D”, J. Comput. Appl. Math. 317 (2017) 685699; doi:10.1016/j.cam.2016.12.004.
[23] Wang, H., “A time-splitting spectral method for coupled Gross–Pitaevskii equations with applications to rotating Bose–Einstein condensates”, J. Comput. Appl. Math. 205 (2007) 88104; doi:10.1016/j.cam.2006.04.042.
[24] Wang, T. C., Guo, B. L. and Xu, Q. B., “Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions”, J. Comput. Phys. 243 (2013) 382399; doi:10.1016/j.jcp.2013.03.007.
[25] Wang, T. C., Jiang, J. P. and Xue, X., “Unconditional and optimal $H^{1}$ error estimate of a Crank–Nicolson finite difference scheme for the Gross–Pitaevskii equation with an angular momentum rotation term”, J. Math. Anal. Appl. 459 (2018) 945958; doi:10.1016/j.jmaa.2017.10.073.
[26] Zhang, F., Vłctor, M., Prez, G. and Luis, V., “Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme”, Appl. Math. Comput. 71 (1995) 165177; doi:10.1016/0096-3003(9400152-T.
[27] Zhou, Y. L., Application of discrete functional analysis to the finite difference methods (International Academic Publishers, Beijing, 1990).
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