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April  2021, 14(2): 303-321. doi: 10.3934/krm.2021006

Shadow Lagrangian dynamics for superfluidity

Patrick Henning 1,2,, and  Anders M. N. Niklasson 3,4,

1. 

Department of Mathematics, Ruhr-University Bochum, DE-44801 Bochum, Germany
2. 

Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
3. 

Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
4. 

Department of Information Technology, Uppsala University, SE-751 05 Uppsala, Sweden

* Corresponding author

Received  June 2020 Revised  December 2020 Published  April 2021 Early access  January 2021

Fund Project: P. Henning acknowledge the support by the Swedish Research Council (grant 2016-03339) and the Göran Gustafsson foundation and A. M. N. Niklasson is most grateful for the hospitality and pleasant environment at the Division of Scientific Computing at the department of Information Technology at Uppsala University, where he stayed during his participation of the development of this work. This work is supported by the U.S. Department of Energy, Office of Basic Energy Sciences (FWP LANLE8AN) and by the U.S. Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy Contract No. 892333218NCA000001. The manuscript has the LA-UR number 19-32663

Motivated by a similar approach for Born-Oppenheimer molecular dynamics, this paper proposes an extended "shadow" Lagrangian density for quantum states of superfluids. The extended Lagrangian contains an additional field variable that is forced to follow the wave function of the quantum state through a rapidly oscillating extended harmonic oscillator. By considering the adiabatic limit for large frequencies of the harmonic oscillator, we can derive the two equations of motions, a Schrödinger-type equation for the quantum state and a wave equation for the extended field variable. The equations are coupled in a nonlinear way, but each equation individually is linear with respect to the variable that it defines. The computational advantage of this new system is that it can be easily discretized using linear time stepping methods, where we propose to use a Crank-Nicolson-type approach for the Schrödinger equation and an extended leapfrog scheme for the wave equation. Furthermore, the difference between the quantum state and the extended field variable defines a consistency error that should go to zero if the frequency tends to infinity. By coupling the time-step size in our discretization to the frequency of the harmonic oscillator we can extract an easily computable consistency error indicator that can be used to estimate the numerical error without additional costs. The findings are illustrated in numerical experiments.

Keywords: Gross-Pitaevskii equation, nonlinear Schrödinger equation, shadow Lagrangians, Hamiltonian systems, conservative schemes, finite element methods.
Mathematics Subject Classification: Primary: 35Q55, 65N30, 81Q05; Secondary: 65N12, 65Y20.
Citation: Patrick Henning, Anders M. N. Niklasson. Shadow Lagrangian dynamics for superfluidity. Kinetic and Related Models, 2021, 14 (2) : 303-321. doi: 10.3934/krm.2021006
References:
[1]

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

[2]

A. Aftalion, Vortices in Bose-Einstein Condensates, Progress in Nonlinear Differential Equations and their Applications, 67. Birkhäuser Boston, Inc., Boston, MA, 2006.

[3]

G. D. Akrivis, V. A. Dougalis and O. A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), 31-53.  doi: 10.1007/BF01385769.

[4]

R. Altmann, P. Henning and D. Peterseim, The J-method for the Gross-Pitaevskii eigenvalue problem, preprint, arXiv: 1908.00333, (2019).

[5]

X. Antoine, W. Bao and C. Besse, 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.

[6]

G. Ariel, J. M. Sanz-Serna and R. Tsai, A multiscale technique for finding slow manifolds of stiff mechanical systems, Multiscale Model. Simul., 10 (2012), 1180-1203.  doi: 10.1137/120861461.

[7]

W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.

[8]

W. Bao and Y. Cai, Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation, Math. Comp., 82 (2013), 99-128.  doi: 10.1090/S0025-5718-2012-02617-2.

[9]

W. Bao and Q. Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), 1674-1697.  doi: 10.1137/S1064827503422956.

[10]

W. Bao, H. Wang and P. A. Markowich, Ground, symmetric and central vortex states in rotating Bose-Einstein condensates, Commun. Math. Sci., 3 (2005), 57-88.  doi: 10.4310/CMS.2005.v3.n1.a5.

[11]

C. Besse, A relaxation scheme for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 42 (2004), 934-952.  doi: 10.1137/S0036142901396521.

[12]

S. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift für Physik, 26 (1924), 178-181.  doi: 10.1007/BF01327326.

[13]

R. Car and M. Parrinello, Unified approach for molecular dynamics and density-functional theory, Phys. Rev. Lett., 55 (1985), 2471. doi: 10.1103/PhysRevLett.55.2471.

[14]

F. Dalfovo, S. Giorgini, L. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Reviews of Modern Physics, 71 (1999), 463-512.  doi: 10.1103/RevModPhys.71.463.

[15]

I. Danaila and P. Kazemi, A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation, SIAM J. Sci. Comput., 32 (2010), 2447-2467.  doi: 10.1137/100782115.

[16]

A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzber. Kgl. Preuss. Akad. Wiss., (1924), 261-267.

[17]

D. L. Feder, A. A. Svidzinsky, A. L. Fetter and C. W. Clark, Anomalous modes drive vortex dynamics in confined Bose-Einstein condensates, Phys. Rev. Lett., 86 (2001), 564-567.  doi: 10.1103/PhysRevLett.86.564.

[18]

A. L. Fetter, Rotating trapped Bose-Einstein condensates, AIP Conference Proceedings, 994 (2008), 98-99.  doi: 10.1063/1.2907762.

[19]

P. Henning and A. Målqvist, The finite element method for the time-dependent Gross-Pitaevskii equation with angular momentum rotation, SIAM J. Numer. Anal., 55 (2017), 923-952.  doi: 10.1137/15M1009172.

[20]

P. Henning and D. Peterseim, Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials, Math. Models Methods Appl. Sci., 27 (2017), 2147-2184.  doi: 10.1142/S0218202517500415.

[21]

P. Henning and D. Peterseim, Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: global convergence and computational efficiency, SIAM J. Numer. Anal., 58 (2020), 1744-1772.  doi: 10.1137/18M1230463.

[22]

P. Henning and J. Wärnegård, Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation, Kinet. Relat. Models, 12 (2019), 1247-1271.  doi: 10.3934/krm.2019048.

[23]

E. Jarlebring, S. Kvaal and W. Michiels, An inverse iteration method for eigenvalue problems with eigenvector nonlinearities, SIAM J. Sci. Comput., 36 (2014), A1978-A2001. doi: 10.1137/130910014.

[24]

O. Karakashian and C. Makridakis, A space-time finite element method for the nonlinear Schrödinger equation: The continuous Galerkin method, SIAM J. Numer. Anal., 36 (1999), 1779-1807.  doi: 10.1137/S0036142997330111.

[25]

E. H. Lieb, R. Seiringer and J. Yngvason, A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Comm. Math. Phys., 224 (2001), 17-31.  doi: 10.1007/s002200100533.

[26]

K. Madison, F. Chevy, V. Bretin and J. Dalibard, Stationary states of a rotating Bose-Einstein condensate: Routes to vortex nucleation, Physical Review Letters, 86 (2001), 4443-4446.  doi: 10.1103/PhysRevLett.86.4443.

[27]

K. Madison, F. Chevy, W. Wohlleben and J. Dalibard, Vortex formation in a stirred Bose-Einstein condensate, Physical Review Letters, 84 (2000), 806-809.  doi: 10.1103/PhysRevLett.84.806.

[28]

M. Matthews, B. Anderson, P. Haljan, D. Hall, C. Wieman and E. Cornell, Vortices in a Bose-Einstein condensate, Physical Review Letters, 83 (1999), 2498-2501.  doi: 10.1142/9789812813787_0077.

[29]

A. M. N. Niklasson, Extended Born-Oppenheimer molecular dynamics, Phys. Rev. Lett., 100 (2008), 123004. doi: 10.1103/PhysRevLett.100.123004.

[30]

A. M. N. Niklasson, Next generation extended Lagrangian first principles molecular dynamics, J. Chem. Phys., 147 (2017), 054103. doi: 10.1063/1.4985893.

[31]

A. M. N. Niklasson and M. J. Cawkwell, Generalized extended Lagrangian Born-Oppenheimer molecular dynamics, J. Chem. Phys., 141 (2014), 164123. doi: 10.1063/1.4898803.

[32]

A. M. N. Niklasson, P. Steneteg, A. Odell, N. Bock, M. Challacombe, C. J. Tymczak, E. Holmstrom, G. Zheng and V. Weber, Extended Lagrangian Born-Oppenheimer molecular dynamics with dissipation, J. Chem. Phys., 130 (2009), 214109. doi: 10.1063/1.3148075.

[33]

A. M. N. Niklasson, C. J. Tymczak and M. Challacombe, Time-reversible ab initio molecular dynamics, J. Chem. Phys., 126 (2007), 144103. doi: 10.1063/1.2715556.

[34] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University Press, Oxford, 2003. 
[35]

P. Pulay and G. Fogarasi, Fock matrix dynamics, Chem. Phys. Lett., 386 (2004), 272-278.  doi: 10.1016/j.cplett.2004.01.069.

[36]

D. K. Remler and P. A. Madden, Molecular dynamics without effective potentials via the car-parrinello approach, Mol. Phys., 70 (1990), 921-966.  doi: 10.1080/00268979000101451.

[37]

J. M. Sanz-Serna, Methods for the numerical solution of the nonlinear Schroedinger equation, Math. Comp., 43 (1984), 21-27.  doi: 10.1090/S0025-5718-1984-0744922-X.

[38]

J. M. Sanz-Serna, Runge-Kutta schemes for Hamiltonian systems, BIT, 28 (1988), 877-883.  doi: 10.1007/BF01954907.

[39]

G. Zheng, A. M. N. Niklasson and M. Karplus, Lagrangian formulation with dissipation of Born-Oppenheimer molecular dynamics using the density-functional tight-binding method, J. Chem. Phys., 135 (2011), 044122. doi: 10.1063/1.3605303.

show all references

References:
[1]

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

[2]

A. Aftalion, Vortices in Bose-Einstein Condensates, Progress in Nonlinear Differential Equations and their Applications, 67. Birkhäuser Boston, Inc., Boston, MA, 2006.

[3]

G. D. Akrivis, V. A. Dougalis and O. A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), 31-53.  doi: 10.1007/BF01385769.

[4]

R. Altmann, P. Henning and D. Peterseim, The J-method for the Gross-Pitaevskii eigenvalue problem, preprint, arXiv: 1908.00333, (2019).

[5]

X. Antoine, W. Bao and C. Besse, 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.

[6]

G. Ariel, J. M. Sanz-Serna and R. Tsai, A multiscale technique for finding slow manifolds of stiff mechanical systems, Multiscale Model. Simul., 10 (2012), 1180-1203.  doi: 10.1137/120861461.

[7]

W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.

[8]

W. Bao and Y. Cai, Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation, Math. Comp., 82 (2013), 99-128.  doi: 10.1090/S0025-5718-2012-02617-2.

[9]

W. Bao and Q. Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), 1674-1697.  doi: 10.1137/S1064827503422956.

[10]

W. Bao, H. Wang and P. A. Markowich, Ground, symmetric and central vortex states in rotating Bose-Einstein condensates, Commun. Math. Sci., 3 (2005), 57-88.  doi: 10.4310/CMS.2005.v3.n1.a5.

[11]

C. Besse, A relaxation scheme for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 42 (2004), 934-952.  doi: 10.1137/S0036142901396521.

[12]

S. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift für Physik, 26 (1924), 178-181.  doi: 10.1007/BF01327326.

[13]

R. Car and M. Parrinello, Unified approach for molecular dynamics and density-functional theory, Phys. Rev. Lett., 55 (1985), 2471. doi: 10.1103/PhysRevLett.55.2471.

[14]

F. Dalfovo, S. Giorgini, L. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Reviews of Modern Physics, 71 (1999), 463-512.  doi: 10.1103/RevModPhys.71.463.

[15]

I. Danaila and P. Kazemi, A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation, SIAM J. Sci. Comput., 32 (2010), 2447-2467.  doi: 10.1137/100782115.

[16]

A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzber. Kgl. Preuss. Akad. Wiss., (1924), 261-267.

[17]

D. L. Feder, A. A. Svidzinsky, A. L. Fetter and C. W. Clark, Anomalous modes drive vortex dynamics in confined Bose-Einstein condensates, Phys. Rev. Lett., 86 (2001), 564-567.  doi: 10.1103/PhysRevLett.86.564.

[18]

A. L. Fetter, Rotating trapped Bose-Einstein condensates, AIP Conference Proceedings, 994 (2008), 98-99.  doi: 10.1063/1.2907762.

[19]

P. Henning and A. Målqvist, The finite element method for the time-dependent Gross-Pitaevskii equation with angular momentum rotation, SIAM J. Numer. Anal., 55 (2017), 923-952.  doi: 10.1137/15M1009172.

[20]

P. Henning and D. Peterseim, Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials, Math. Models Methods Appl. Sci., 27 (2017), 2147-2184.  doi: 10.1142/S0218202517500415.

[21]

P. Henning and D. Peterseim, Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: global convergence and computational efficiency, SIAM J. Numer. Anal., 58 (2020), 1744-1772.  doi: 10.1137/18M1230463.

[22]

P. Henning and J. Wärnegård, Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation, Kinet. Relat. Models, 12 (2019), 1247-1271.  doi: 10.3934/krm.2019048.

[23]

E. Jarlebring, S. Kvaal and W. Michiels, An inverse iteration method for eigenvalue problems with eigenvector nonlinearities, SIAM J. Sci. Comput., 36 (2014), A1978-A2001. doi: 10.1137/130910014.

[24]

O. Karakashian and C. Makridakis, A space-time finite element method for the nonlinear Schrödinger equation: The continuous Galerkin method, SIAM J. Numer. Anal., 36 (1999), 1779-1807.  doi: 10.1137/S0036142997330111.

[25]

E. H. Lieb, R. Seiringer and J. Yngvason, A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Comm. Math. Phys., 224 (2001), 17-31.  doi: 10.1007/s002200100533.

[26]

K. Madison, F. Chevy, V. Bretin and J. Dalibard, Stationary states of a rotating Bose-Einstein condensate: Routes to vortex nucleation, Physical Review Letters, 86 (2001), 4443-4446.  doi: 10.1103/PhysRevLett.86.4443.

[27]

K. Madison, F. Chevy, W. Wohlleben and J. Dalibard, Vortex formation in a stirred Bose-Einstein condensate, Physical Review Letters, 84 (2000), 806-809.  doi: 10.1103/PhysRevLett.84.806.

[28]

M. Matthews, B. Anderson, P. Haljan, D. Hall, C. Wieman and E. Cornell, Vortices in a Bose-Einstein condensate, Physical Review Letters, 83 (1999), 2498-2501.  doi: 10.1142/9789812813787_0077.

[29]

A. M. N. Niklasson, Extended Born-Oppenheimer molecular dynamics, Phys. Rev. Lett., 100 (2008), 123004. doi: 10.1103/PhysRevLett.100.123004.

[30]

A. M. N. Niklasson, Next generation extended Lagrangian first principles molecular dynamics, J. Chem. Phys., 147 (2017), 054103. doi: 10.1063/1.4985893.

[31]

A. M. N. Niklasson and M. J. Cawkwell, Generalized extended Lagrangian Born-Oppenheimer molecular dynamics, J. Chem. Phys., 141 (2014), 164123. doi: 10.1063/1.4898803.

[32]

A. M. N. Niklasson, P. Steneteg, A. Odell, N. Bock, M. Challacombe, C. J. Tymczak, E. Holmstrom, G. Zheng and V. Weber, Extended Lagrangian Born-Oppenheimer molecular dynamics with dissipation, J. Chem. Phys., 130 (2009), 214109. doi: 10.1063/1.3148075.

[33]

A. M. N. Niklasson, C. J. Tymczak and M. Challacombe, Time-reversible ab initio molecular dynamics, J. Chem. Phys., 126 (2007), 144103. doi: 10.1063/1.2715556.

[34] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University Press, Oxford, 2003. 
[35]

P. Pulay and G. Fogarasi, Fock matrix dynamics, Chem. Phys. Lett., 386 (2004), 272-278.  doi: 10.1016/j.cplett.2004.01.069.

[36]

D. K. Remler and P. A. Madden, Molecular dynamics without effective potentials via the car-parrinello approach, Mol. Phys., 70 (1990), 921-966.  doi: 10.1080/00268979000101451.

[37]

J. M. Sanz-Serna, Methods for the numerical solution of the nonlinear Schroedinger equation, Math. Comp., 43 (1984), 21-27.  doi: 10.1090/S0025-5718-1984-0744922-X.

[38]

J. M. Sanz-Serna, Runge-Kutta schemes for Hamiltonian systems, BIT, 28 (1988), 877-883.  doi: 10.1007/BF01954907.

[39]

G. Zheng, A. M. N. Niklasson and M. Karplus, Lagrangian formulation with dissipation of Born-Oppenheimer molecular dynamics using the density-functional tight-binding method, J. Chem. Phys., 135 (2011), 044122. doi: 10.1063/1.3605303.

Figure 1.  Model problem 1. Left: ground state density $ |u_0|^2 $ at $ t = 0 $. Middle: exact density $ |u(t)|^2 $ at time $ t = 2 $ Right: exact density at final computing time $ t = 4 $
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Figure 2.  Model Problem 1. Comparison of accuracies for the Dissipative Shadow Lagrangian Method with different dissipation orders $K$. Notably, the accuracies for $K$ = 3, 4, 5, 6 are extremely close to each other
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Figure 3.  Model Problem 1. Left: Comparison of the ${H^1}$-errors for DS-K$5$, the classical Crank-Nicolson scheme and the Besserelaxation scheme. Right: comparison of the exact ${H^1}$-error for DS-K$5$ with the estimated error using the consistency indicator $\left|E\left(\psi^{N}\right)-E\left(\phi^{N}\right)\right|$
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Figure 4.  Model Problem 1. Results obtained for DS-K$5$ and three different step sizes $\tau $. Left: Variations of the energy E (cf. $(13)$) over time. Right: Variations of the mass $m\left(\psi^{n}\right) = \left\|\psi^{n}\right\|_{L^{2}(U)}$ over time
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Figure 5.  Model problem 2. Left: potential $ V_0 $ as given by $(14)$. Middle: ground state density $ |u_0|^2 $, i.e., $ |u(t)|^2 $ at $ t = 0 $. Right: exact density $ |u(t)|^2 $ at final computing time $ t = 1 $
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Figure 6.  Model Problem 2. Comparison of the accuracies for DS-K$5$, the classical Crank-Nicolson scheme and the Besse-relaxation scheme. Left: comparison of ${L^2}$-errors. Right: comparison of ${H^1}$- errors with visibly reduced convergence rates
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Figure 7.  Model Problem 2. Left: Consistency error $\psi^{N}-\phi^{N}$ in the ${L^2}$- and the ${H^1}$-norm for various step sizes $\tau $. Right: Comparison of the exact ${H^1}$-error for DS-K$5$ with the estimated error using the consistency indicator $\left|E\left(\psi^{N}\right)-E\left(\phi^{N}\right)\right|$
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Figure 8.  Model Problem 2. Results obtained for DS-K$5$ and three different step sizes $\tau $. Left: Variations of the energy E (cf. $(15)$) over time. Right: Variations of the mass $m\left(\psi^{n}\right) = \left\|\psi^{n}\right\|_{L^{2}(U)}$ over time
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