# zbMATH — the first resource for mathematics

A new linear and conservative finite difference scheme for the Gross-Pitaevskii equation with angular momentum rotation. (English) Zbl 1415.65189
Summary: 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 $$O(h^4+\tau^{2})$$ with time step $$\tau$$ and mesh size $$h$$. Numerical experiments have been carried out to show the efficiency and accuracy of our new method.
##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
GPELab
Full Text:
##### References:
 [1] Antoine, X.; Bao, W.; Besse, C., Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Comm., 184, 2621-2633, (2013) · Zbl 1344.35130 [2] Antoine, X.; Duboscq, R., GPELab, a Matlab toolbox to solve Gross-Pitaevskii equations I: Computation of stationary solutions, Comput. Phys. Comm., 185, 2969-2991, (2014) · Zbl 1348.35003 [3] Bao, W.; Cai, Y., Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation, Math. Comp., 82, 99-128, (2013) · Zbl 1264.65146 [4] Bao, W.; Du, Q., Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25, 1674-1697, (2004) · Zbl 1061.82025 [5] Browder, F. E.; Finn, R., Existence and uniqueness theorems for solutions of nonlinear boundary value problems, Applications of nonlinear partial differential equations, Volume 17 of, 24-49, (1965), American Mathematical Society: American Mathematical Society, Providence [6] Castin, Y.; Dum, R., Bose-Einstein condensates with vortices in rotating traps, Eur. Phys. J. D, 7, 399-412, (1999) [7] Chang, Q.; Jia, E.; Sun, W., Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys., 148, 397-415, (1999) · Zbl 0923.65059 [8] Cloot, A.; Herbst, B. M.; Weideman, J. A. C., A numerical study of the nonlinear Schrödinger equation involving quintic terms, J. Comput. Phys., 86, 127-146, (1990) · Zbl 0685.65110 [9] Dalfovo, F.; Giorgini, S., Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71, 463-512, (1999) [10] Gong, Y. Z.; Wang, Q.; Wang, Y. S.; Cai, J. X., A conservative Fourier pseudospectral method for the nonlinear Schrödinger equation, J. Comput. Phys., 328, 354-370, (2017) · Zbl 1406.35356 [11] Gray, R. [12] Guo, B. Y., The convergence of numerical method for nonlinear Schrödinger equation, J. Comput. Math., 4, 121-130, (1986) · Zbl 0598.65092 [13] Hao, C. C.; Hsiao, L.; Li, H. L., Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term, Math. Methods Appl. Sci., 31, 655-664, (2008) · Zbl 1132.35476 [14] Henning, P.; Malqvist, A., The finite element method for the time-dependent Gross-Pitaevskii equation with angular momentum rotation, SIAM J. Numer. Anal., 55, 923-952, (2017) · Zbl 1362.65105 [15] Lees, M., Approximate solutions of parabolic equations, J. Soc. Ind. Appl. Math., 7, 167-183, (1959) · Zbl 0086.32801 [16] Liao, H. L.; Sun, Z. Z., Error estimate of fourth-order compact scheme for linear Schrödinger equations, SIAM J. Numer. Anal., 47, 4381-4401, (2010) · Zbl 1208.65130 [17] Lieb, E. H.; Seiringer, R., Derivation of the Gross-Pitaevskii equation for rotating Bose gases, Commun. Math. Phys., 264, 505-537, (2006) · Zbl 1233.82004 [18] Madison, K. W.; Chevy, F.; Wohlleben, W.; Dalibard, J., Vortex formation in a stirred Bose-Einstein condensate, Phys. Rev. Lett., 84, 806-809, (2000) [19] Matthews, M. R.; Anderson, B. P.; Haljan, P. C.; Hall, D. S.; Wieman, C. E.; Cornell, E. A., Vortices in a Bose-Einstein condensate, Phys. Rev. Lett., 83, 2498-2501, (1999) [20] Pitaevskii, L.; Stringary, S., Bose-Einstein condensation, Volume 116 of, (2003), Clarendon Press: Clarendon Press, Oxford [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, 1595-1619, (2003) · Zbl 1053.65085 [22] Sun, W. W.; Wang, J. L., Optimal error analysis of Crank-Nicolson schemes for a coupled nonlinear Schrödinger system in 3D, J. Comput. Appl. Math., 317, 685-699, (2017) · Zbl 1357.65148 [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, 88-104, (2007) · Zbl 1118.65112 [24] Wang, T. C.; Guo, B. L.; Xu, Q. B., Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions, J. Comput. Phys., 243, 382-399, (2013) · Zbl 1349.65347 [25] Wang, T. C.; Jiang, J. P.; 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, 945-958, (2018) · Zbl 1379.65066 [26] Zhang, F.; Vłctor, M.; Prez, G.; Luis, V., Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme, Appl. Math. Comput., 71, 165-177, (1995) [27] Zhou, Y. L., Application of discrete functional analysis to the finite difference methods, (1990), International Academic Publishers: International Academic Publishers, Beijing
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.