American Institute of Mathematical Sciences

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December  2019, 6(2): 147-169. doi: 10.3934/jcd.2019008

Efficient time integration methods for Gross-Pitaevskii equations with rotation term

Philipp Bader 1, , Sergio Blanes 2, , Fernando Casas 3, and  Mechthild Thalhammer 4,

1. 

Universitat Jaume I, Departament de Matemàtiques, 12071 Castellón, Spain
2. 

Universitat Politècnica de València, Instituto Universitario de Matemática Multidisciplinar, 46022 Valencia, Spain
3. 

Universitat Jaume I, IMAC and Departament de Matemàtiques, 12071 Castellón, Spain
4. 

Leopold-Franzens-Universität Innsbruck, Institut für Mathematik, 6020 Innsbruck, Austria

Received  March 2019 Revised  August 2019 Published  November 2019

The objective of this work is the introduction and investigation of favourable time integration methods for the Gross-Pitaevskii equation with rotation term. Employing a reformulation in rotating Lagrangian coordinates, the equation takes the form of a nonlinear Schrödinger equation involving a space-time-dependent potential. A natural approach that combines commutator-free quasi-Magnus exponential integrators with operator splitting methods and Fourier spectral space discretisations is proposed. Furthermore, the special structure of the Hamilton operator permits the design of specifically tailored schemes. Numerical experiments confirm the good performance of the resulting exponential integrators.

Keywords: Nonlinear Schrödinger equations, Gross-Pitaevskii equations, exponential integrators, Magnus integrators, commutator-free quasi-Magnus integrators, spectral methods, fast Fourier transform.
Mathematics Subject Classification: Primary: 35Q55, 65L20, 65M12; Secondary: 58G35.
Citation: Philipp Bader, Sergio Blanes, Fernando Casas, Mechthild Thalhammer. Efficient time integration methods for Gross-Pitaevskii equations with rotation term. Journal of Computational Dynamics, 2019, 6 (2) : 147-169. doi: 10.3934/jcd.2019008
References:
[1]

J. R. Abo-Shaeer, C. Raman, J. M. Vogels and W. Ketterle, Observation of vortex lattices in Bose-Einstein condensates, Science, 292 (2001), 476-479.  doi: 10.1126/science.1060182.  Google Scholar

[2]

A. Alvermann and H. Fehske, High-order commutator-free exponential time-propagation of driven quantum systems, J. Comput. Phys., 230 (2011), 5930-5956.  doi: 10.1016/j.jcp.2011.04.006.  Google Scholar

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A. Alvermann, H. Fehske and P. B. Littlewood, Numerical time propagation of quantum systems in radiation fields, New. J. Phys., 14 (2012), 22pp.  doi: 10.1088/1367-2630/14/10/105008.  Google Scholar

[4]

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[5]

X. Antoine, C. Besse and W. Bao, Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun., 184 (2013), 2621-2633.  doi: 10.1016/j.cpc.2013.07.012.  Google Scholar

[6]

P. Bader, Geometric integrators for Schrödinger equations, Ph.D thesis, Universitat Politécnica de Valencia, 2014. doi: 10.4995/Thesis/10251/38716.  Google Scholar

[7]

P. Bader, Fourier-splitting methods for the dynamics of rotating Bose-Einstein condensates, J. Comput. Appl. Math., 336 (2018), 267-280.  doi: 10.1016/j.cam.2017.12.038.  Google Scholar

[8]

P. Bader, S. Blanes and N. Kopylov, Exponential propagators for the Schrödinger equation with a time-dependent potential, J. Chem. Phys., 148 (2018).  doi: 10.1063/1.5036838.  Google Scholar

[9]

W. Bao, H. Li and J. Shen, A generalized Laguerre-Fourier-Hermite pseudospectral method for computing the dynamics of rotating Bose-Einstein condensates, SIAM J. Sci. Comput., 31 (2009), 3685-3711.  doi: 10.1137/080739811.  Google Scholar

[10]

W. Bao, D. Marahrens, Q. Tang and Y. Zhang, A simple and efficient numerical method for computing the dynamics of rotating Bose-Einstein condensates via rotating Lagrangian coordinates, SIAM J. Sci. Comput., 35 (2013), A2671-A2695.  doi: 10.1137/130911111.  Google Scholar

[11]

C. Besse, G. Dujardin and I. Lacroix-Violet, High order exponential integrators for nonlinear Schrödinger equations with application to rotating Bose-Einstein condensates, SIAM J. Numer. Anal., 55 (2017), 1387-1411.  doi: 10.1137/15M1029047.  Google Scholar

[12]

S. Blanes, Time-average on the numerical integration of nonautonomous differential equations, SIAM J. Numer. Anal., 56 (2018), 2513-2536.  doi: 10.1137/17M1156150.  Google Scholar

[13]

S. Blanes, F. Casas, C. González and M. Thalhammer, Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrödinger equations, IMA J. Numer. Anal., 38 (2018), 743-778.  doi: 10.1093/imanum/drx012.  Google Scholar

[14]

S. Blanes, F. Casas, J. A. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Rep., 470 (2009), 151-238.  doi: 10.1016/j.physrep.2008.11.001.  Google Scholar

[15]

S. Blanes, F. Casas and M. Thalhammer, High-order commutator-free quasi-Magnus exponential integrators for non-autonomous linear evolution equations, Comput. Phys. Commun., 220 (2017), 243-262.  doi: 10.1016/j.cpc.2017.07.016.  Google Scholar

[16]

S. Blanes, F. Casas and M. Thalhammer, Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of parabolic type, IMA J. Numer. Anal., 38 (2018), 743-778.  doi: 10.1093/imanum/drx012.  Google Scholar

[17]

S. Blanes, F. Diele, C. Marangi and S. Ragni, Splitting and composition methods for explicit time dependence in separable dynamical systems, J. Comput. Appl. Math., 235 (2010), 646-659.  doi: 10.1016/j.cam.2010.06.018.  Google Scholar

[18]

S. Blanes and P. C. Moan, Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nyström methods, J. Comput. Appl. Math., 142 (2002), 313-330.  doi: 10.1016/S0377-0427(01)00492-7.  Google Scholar

[19]

C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet, Evidence of Bose-Einstein condensation in an atomic gas with attractive interaction, Phys. Rev. Lett., 75 (1995), 1687-1690.  doi: 10.1103/PhysRevLett.75.1687.  Google Scholar

[20]

I. Danaila and B. Protas, Computation of ground states of the Gross-Pitaevskii functional via Riemannian optimization, SIAM J. Sci. Comput., 39 (2017), B1102-B1129.  doi: 10.1137/17M1121974.  Google Scholar

[21]

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1995), 3969-3973.  doi: 10.1103/PhysRevLett.75.3969.  Google Scholar

[22]

H. Hofstätter, O. Koch and M. Thalhammer, Convergence analysis of high-order time-splitting pseudo-spectral methods for Gross-Pitaevskii equations with rotation term, Numer. Math., 127 (2014), 315-364.  doi: 10.1007/s00211-013-0586-9.  Google Scholar

[23]

A. Iserles and S. P. Nørsett, On the solution of linear differential equations in Lie groups, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 983-1019.  doi: 10.1098/rsta.1999.0362.  Google Scholar

[24]

A. Iserles and G. R. W. Quispel, Why Geometric Numerical Integration?, Discrete Mechanics, Geometric Integration and Lie-Butcher Series, 267, Springer, Cham, 2018, 1-28. doi: 10.1007/978-3-030-01397-4_1.  Google Scholar

[25]

K. W. Madison, F. Chevy, V. Bretin and J. Dalibard, Stationary states of a rotating Bose-Einstein condensates: Routes to vortex nucleation, Phys. Rev. Lett., 86 (2001), 4443-4446.  doi: 10.1103/PhysRevLett.86.4443.  Google Scholar

[26]

W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math., 7 (1954), 649-673.  doi: 10.1002/cpa.3160070404.  Google Scholar

[27]

R. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numer., 11 (2002), 341-434.  doi: 10.1017/S0962492902000053.  Google Scholar

[28]

H. Munthe-Kaas and B. Owren, Computations in a free Lie algebra, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 957-981.  doi: 10.1098/rsta.1999.0361.  Google Scholar

[29]

M. Thalhammer, A fourth-order commutator-free exponential integrator for nonautonomous differential equations, SIAM J. Numer. Anal., 44 (2006), 851-864.  doi: 10.1137/05063042.  Google Scholar

[30]

M. Thalhammer, Convergence analysis of high-order time-splitting pseudospectral methods for nonlinear Schrödinger equations, SIAM J. Numer. Anal., 50 (2012), 3231-3258.  doi: 10.1137/120866373.  Google Scholar

[31]

M. Thalhammer and J. Abhau, A numerical study of adaptive space and time discretisations for Gross-Pitaevskii equations, J. Comput. Phys., 231 (2012), 6665-6681.  doi: 10.1016/j.jcp.2012.05.031.  Google Scholar

[32]

R. Zeng and Y. Zhang, Efficiently computing vortex lattices in rapid rotating Bose-Einstein condensates, Comput. Phys. Comm., 180 (2009), 854-860.  doi: 10.1016/j.cpc.2008.12.003.  Google Scholar

show all references

References:
[1]

J. R. Abo-Shaeer, C. Raman, J. M. Vogels and W. Ketterle, Observation of vortex lattices in Bose-Einstein condensates, Science, 292 (2001), 476-479.  doi: 10.1126/science.1060182.  Google Scholar

[2]

A. Alvermann and H. Fehske, High-order commutator-free exponential time-propagation of driven quantum systems, J. Comput. Phys., 230 (2011), 5930-5956.  doi: 10.1016/j.jcp.2011.04.006.  Google Scholar

[3]

A. Alvermann, H. Fehske and P. B. Littlewood, Numerical time propagation of quantum systems in radiation fields, New. J. Phys., 14 (2012), 22pp.  doi: 10.1088/1367-2630/14/10/105008.  Google Scholar

[4]

M. H. Anderson, J. R. Ensher, M. R. Matthewa, C. E. Wieman and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269 (1995), 198-201.  doi: 10.1126/science.269.5221.198.  Google Scholar

[5]

X. Antoine, C. Besse and W. Bao, Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun., 184 (2013), 2621-2633.  doi: 10.1016/j.cpc.2013.07.012.  Google Scholar

[6]

P. Bader, Geometric integrators for Schrödinger equations, Ph.D thesis, Universitat Politécnica de Valencia, 2014. doi: 10.4995/Thesis/10251/38716.  Google Scholar

[7]

P. Bader, Fourier-splitting methods for the dynamics of rotating Bose-Einstein condensates, J. Comput. Appl. Math., 336 (2018), 267-280.  doi: 10.1016/j.cam.2017.12.038.  Google Scholar

[8]

P. Bader, S. Blanes and N. Kopylov, Exponential propagators for the Schrödinger equation with a time-dependent potential, J. Chem. Phys., 148 (2018).  doi: 10.1063/1.5036838.  Google Scholar

[9]

W. Bao, H. Li and J. Shen, A generalized Laguerre-Fourier-Hermite pseudospectral method for computing the dynamics of rotating Bose-Einstein condensates, SIAM J. Sci. Comput., 31 (2009), 3685-3711.  doi: 10.1137/080739811.  Google Scholar

[10]

W. Bao, D. Marahrens, Q. Tang and Y. Zhang, A simple and efficient numerical method for computing the dynamics of rotating Bose-Einstein condensates via rotating Lagrangian coordinates, SIAM J. Sci. Comput., 35 (2013), A2671-A2695.  doi: 10.1137/130911111.  Google Scholar

[11]

C. Besse, G. Dujardin and I. Lacroix-Violet, High order exponential integrators for nonlinear Schrödinger equations with application to rotating Bose-Einstein condensates, SIAM J. Numer. Anal., 55 (2017), 1387-1411.  doi: 10.1137/15M1029047.  Google Scholar

[12]

S. Blanes, Time-average on the numerical integration of nonautonomous differential equations, SIAM J. Numer. Anal., 56 (2018), 2513-2536.  doi: 10.1137/17M1156150.  Google Scholar

[13]

S. Blanes, F. Casas, C. González and M. Thalhammer, Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrödinger equations, IMA J. Numer. Anal., 38 (2018), 743-778.  doi: 10.1093/imanum/drx012.  Google Scholar

[14]

S. Blanes, F. Casas, J. A. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Rep., 470 (2009), 151-238.  doi: 10.1016/j.physrep.2008.11.001.  Google Scholar

[15]

S. Blanes, F. Casas and M. Thalhammer, High-order commutator-free quasi-Magnus exponential integrators for non-autonomous linear evolution equations, Comput. Phys. Commun., 220 (2017), 243-262.  doi: 10.1016/j.cpc.2017.07.016.  Google Scholar

[16]

S. Blanes, F. Casas and M. Thalhammer, Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of parabolic type, IMA J. Numer. Anal., 38 (2018), 743-778.  doi: 10.1093/imanum/drx012.  Google Scholar

[17]

S. Blanes, F. Diele, C. Marangi and S. Ragni, Splitting and composition methods for explicit time dependence in separable dynamical systems, J. Comput. Appl. Math., 235 (2010), 646-659.  doi: 10.1016/j.cam.2010.06.018.  Google Scholar

[18]

S. Blanes and P. C. Moan, Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nyström methods, J. Comput. Appl. Math., 142 (2002), 313-330.  doi: 10.1016/S0377-0427(01)00492-7.  Google Scholar

[19]

C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet, Evidence of Bose-Einstein condensation in an atomic gas with attractive interaction, Phys. Rev. Lett., 75 (1995), 1687-1690.  doi: 10.1103/PhysRevLett.75.1687.  Google Scholar

[20]

I. Danaila and B. Protas, Computation of ground states of the Gross-Pitaevskii functional via Riemannian optimization, SIAM J. Sci. Comput., 39 (2017), B1102-B1129.  doi: 10.1137/17M1121974.  Google Scholar

[21]

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1995), 3969-3973.  doi: 10.1103/PhysRevLett.75.3969.  Google Scholar

[22]

H. Hofstätter, O. Koch and M. Thalhammer, Convergence analysis of high-order time-splitting pseudo-spectral methods for Gross-Pitaevskii equations with rotation term, Numer. Math., 127 (2014), 315-364.  doi: 10.1007/s00211-013-0586-9.  Google Scholar

[23]

A. Iserles and S. P. Nørsett, On the solution of linear differential equations in Lie groups, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 983-1019.  doi: 10.1098/rsta.1999.0362.  Google Scholar

[24]

A. Iserles and G. R. W. Quispel, Why Geometric Numerical Integration?, Discrete Mechanics, Geometric Integration and Lie-Butcher Series, 267, Springer, Cham, 2018, 1-28. doi: 10.1007/978-3-030-01397-4_1.  Google Scholar

[25]

K. W. Madison, F. Chevy, V. Bretin and J. Dalibard, Stationary states of a rotating Bose-Einstein condensates: Routes to vortex nucleation, Phys. Rev. Lett., 86 (2001), 4443-4446.  doi: 10.1103/PhysRevLett.86.4443.  Google Scholar

[26]

W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math., 7 (1954), 649-673.  doi: 10.1002/cpa.3160070404.  Google Scholar

[27]

R. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numer., 11 (2002), 341-434.  doi: 10.1017/S0962492902000053.  Google Scholar

[28]

H. Munthe-Kaas and B. Owren, Computations in a free Lie algebra, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 957-981.  doi: 10.1098/rsta.1999.0361.  Google Scholar

[29]

M. Thalhammer, A fourth-order commutator-free exponential integrator for nonautonomous differential equations, SIAM J. Numer. Anal., 44 (2006), 851-864.  doi: 10.1137/05063042.  Google Scholar

[30]

M. Thalhammer, Convergence analysis of high-order time-splitting pseudospectral methods for nonlinear Schrödinger equations, SIAM J. Numer. Anal., 50 (2012), 3231-3258.  doi: 10.1137/120866373.  Google Scholar

[31]

M. Thalhammer and J. Abhau, A numerical study of adaptive space and time discretisations for Gross-Pitaevskii equations, J. Comput. Phys., 231 (2012), 6665-6681.  doi: 10.1016/j.jcp.2012.05.031.  Google Scholar

[32]

R. Zeng and Y. Zhang, Efficiently computing vortex lattices in rapid rotating Bose-Einstein condensates, Comput. Phys. Comm., 180 (2009), 854-860.  doi: 10.1016/j.cpc.2008.12.003.  Google Scholar

Figure 1.  Overview of exponential time integration methods. Commutator-free quasi-Magnus exponential integrators (CFQM, order $ p $, number of compositions $ J $, number of quadrature nodes $ K $) or modified commutator-free quasi-Magnus exponential integrator (Scheme BBK, order $ p = 6 $), respectively, combined with operator splitting methods (order $ p $, number of compositions $ s $)
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Figure 2.  Time integration of non-autonomous linear test equation in two (left) and three (right) space dimensions by quasi-Magnus exponential integrators and operator splitting methods, see (8), (21), and Figure 1. Global errors versus time stepsizes (first row) or number of fast Fourier transforms and inverse fast Fourier transforms (second row), respectively
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Figure 3.  Time integration of non-autonomous nonlinear test equation in two (left) and three (right) space dimensions with $ \vartheta = 1 $ by quasi-Magnus exponential integrators and operator splitting methods, see (8), (21), and Figure 1. Global errors versus time stepsizes (first row) or number of fast Fourier transforms and inverse fast Fourier transforms (second row), respectively
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Figure 4.  Time integration of non-autonomous nonlinear test equation in two (left) and three (right) space dimensions with $ \vartheta = 10 $ by quasi-Magnus exponential integrators and operator splitting methods, see (8), (21), and Figure 1. Global errors versus time stepsizes (first row) or number of fast Fourier transforms and inverse fast Fourier transforms (second row), respectively
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Figure 5.  Time integration of two-dimensional rotational Gross-Pitaevskii equation (22) by sixth-order modified CFQM exponential integrator (20) and optimised sixth-order splitting method proposed in [18]. Solution profile $ |{\psi(x, t)}|^2 $ displayed for spatial section $ x \in [- 5, 5]^2 $ and time $ t = 15 $. Consistent results obtained by time-splitting generalised-Laguerre-Fourier spectral methods [22,Eq. (7)-(8),Fig. 2]
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