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Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions

1 Center for Theoretical Physics of Complex Systems, Institute for Basic Sciences, Daejeon, Korea
2 Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701, Cape Town, South Africa
3 Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany

$^\dagger$This contribution is part of the Special Issue: Hamiltonian Lattice Dynamics
  Guest Editors: Simone Paleari; Tiziano Penati
  Link: http://www.aimspress.com/newsinfo/1165.html

Special Issues: Hamiltonian Lattice Dynamics

We investigate the computational performance of various numerical methods for the integration of the equations of motion and the variational equations for some typical classical many-body models of condensed matter physics: the Fermi-Pasta-Ulam-Tsingou (FPUT) chain and the one- and two-dimensional disordered, discrete nonlinear Schrödinger equations (DDNLS). In our analysis we consider methods based on Taylor series expansion, Runge-Kutta discretization and symplectic transformations. The latter have the ability to exactly preserve the symplectic structure of Hamiltonian systems, which results in keeping bounded the error of the system’s computed total energy. We perform extensive numerical simulations for several initial conditions of the studied models and compare the numerical efficiency of the used integrators by testing their ability to accurately reproduce characteristics of the systems’ dynamics and quantify their chaoticity through the computation of the maximum Lyapunov exponent. We also report the expressions of the implemented symplectic schemes and provide the explicit forms of the used differential operators. Among the tested numerical schemes the symplectic integrators $ABA864$ and $SRKN^a_{14}$ exhibit the best performance, respectively for moderate and high accuracy levels in the case of the FPUT chain, while for the DDNLS models $s9\mathcal{ABC}6$ and $s11\mathcal{ABC}6$ (moderate accuracy), along with $s17\mathcal{ABC}8$ and $s19\mathcal{ABC}8$ (high accuracy) proved to be the most efficient schemes.
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References

1. Fermi E, Pasta P, Ulam S, et al.(1955) Studies of the nonlinear problems, Los Alamos Report LA-1940.

2. Ford J (1992) The Fermi-Pasta-Ulam problem: Paradox turns discovery. Phys Rep 213: 271–310.    

3. Campbell DK, Rosenau P, Zaslavsky GM (2005) Introduction: The Fermi-Pasta-Ulam problem: The first fifty years. Chaos 15: 015101.    

4. Lepri S, Livi R, Politi A (2003) Universality of anomalous one-dimensional heat conductivity. Phys Rev E 68: 067102.    

5. Lepri S, Livi R, Politi A (2005) Studies of thermal conductivity in Fermi-Pasta-Ulam-like lattices. Chaos 15: 015118.    

6. Antonopoulos C, Bountis T, Skokos Ch (2006) Chaotic dynamics of N-degree of freedom Hamiltonian systems. Int J Bifurcation Chaos 16: 1777–1793.    

7. Zabusky NJ, Kruskal MD (1965) Interaction of "solitons" in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15: 240–243.

8. Zabusky NJ, Deem GS (1967) Dynamics of nonlinear lattices I. Localized optical excitations, acoustic radiation, and strong nonlinear behavior. J Comput Phys 2: 126–153.

9. Izrailev FM, Chirikov BV (1966) Statistical properties of a nonlinear string. Sov Phys Dokl 11: 30pages.

10. Paleari S, Penati T (2008) Numerical methods and results in the FPU problem. Lect Notes Phys 728: 239–282.    

11. Anderson PW (1958) Absence of diffusion in certain random lattices. Phys Rev 109: 1492–1505.    

12. Abraham E, Anderson PW, Licciardello DC, et al. (1979) Scaling theory of localization: Absence of quantum diffusion in two dimensions. Phys Rev Lett 42: 673–676.    

13. Shepelyansky DL (1993) Delocalization of quantum chaos by weak nonlinearity. Phys Rev Lett 70: 1787–1790.    

14. Molina MI (1998) Transport of localized and extended excitations in a nonlinear Anderson model. Phys Rev B 58: 12547–12550.    

15. Clément D, Varon AF, Hugbart M, et al. (2005) Suppression of transport of an interacting elongated Bose-Einstein condensate in a random potential. Phys Rev Lett 95: 170409.    

16. Fort C, Fallani L, Guarrera V, et al. (2005) Effect of optical disorder and single defects on the expansion of a Bose-Einstein condensate in a one-dimensional waveguide. Phys Rev Lett 95: 170410.    

17. Schwartz T, Bartal G, Fishman S, et al. (2007) Transport and Anderson localization in disordered two-dimensional photonic lattices. Nature 446: 52–55.    

18. Lahini Y, Avidan A, Pozzi F, et al. (2008) Anderson localization and nonlinearity in one-dimensional disordered photonic lattices. Phys Rev Lett 100: 013906.    

19. Billy J, Josse V, Zuo Z, et al. (2008) Direct observation of Anderson localization of matter waves in a controlled disorder. Nature 453: 891–894.    

20. Roati JG, D'Errico C, Fallani L, et al. (2008) Anderson localization of a non-interacting Bose–Einstein condensate. Nature 453: 895–898.    

21. Fallani L, Fort C, Inguscio M (2008) Bose-Einstein condensates in disordered potentials. Adv At Mol Opt Phys 56: 119–160.    

22. Flach S, Krimer DO, Skokos Ch (2009) Universal spreading of wave packets in disordered nonlinear systems. Phys Rev Lett 102: 024101.    

23. Vicencio RA, Flach S (2009) Control of wave packet spreading in nonlinear finite disordered lattices. Phys Rev E 79: 016217.    

24. Skokos Ch, Krimer DO, Komineas S, et al. (2009) Delocalization of wave packets in disordered nonlinear chains. Phys Rev E 79: 056211.    

25. Skokos Ch, Flach S (2010) Spreading of wave packets in disordered systems with tunable nonlinearity. Phys Rev E 82: 016208.    

26. Laptyeva TV, Bodyfelt JD, Krimer DO, et al. (2010) The crossover from strong to weak chaos for nonlinear waves in disordered systems. EPL 91: 30001.    

27. Modugno G (2010) Anderson localization in Bose-Einstein condensates. Rep Prog Phys 73: 102401.    

28. Bodyfelt JD, Laptyeva TV, Gligoric G, et al. (2011) Wave interactions in localizing media - a coin with many faces, Int J Bifurcat Chaos 21: 2107.

29. Bodyfelt JD, Laptyeva TV, Skokos Ch, et al. (2011) Nonlinear waves in disordered chains: Probing the limits of chaos and spreading. Phys Rev E 84: 016205.    

30. Laptyeva TV, Bodyfelt JD, Flach S (2012) Subdiffusion of nonlinear waves in two-dimensional disordered lattices. EPL 98: 60002.    

31. Pikovsky AS, Shepelyansky DL (2008) Destruction of Anderson localization by a weak nonlinearity. Phys Rev Lett 100: 094101.    

32. Skokos Ch, Gkolias I, Flach S (2013) Nonequilibrium chaos of disordered nonlinear waves. Phys Rev Lett 111: 064101.    

33. Senyangue B, Many Manda B, Skokos Ch (2018) Characteristics of chaos evolution in one- dimensional disordered nonlinear lattices. Phys Rev E 98: 052229.    

34. Kati Y, Yu X, Flach S (2019) Density resolved wave packet spreading in disordered Gross-Pitaevskii lattices. In preparation.

35. Benettin G, Galgani L, Giorgilli A, et al. (1980) Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 2: Numerical application. Meccanica 15: 21–30.

36. Benettin G, Galgani L, Giorgilli A, et al. (1980) Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory. Meccanica 15: 9–20.    

37. Skokos Ch (2010) The Lyapunov characteristic exponents and their computation. Lect Notes Phys 790: 63–135.    

38. Mulansky M, Ahnert K, Pikovsky A, et al. (2009) Dynamical thermalization of disordered nonlinear lattices. Phys Rev E 80: 056212.    

39. Sales MO, Dias WS, Neto AR, et al. (2018) Sub-diffusive spreading and anomalous localization in a 2D Anderson model with off-diagonal nonlinearity. Solid State Commun 270: 6–11.    

40. Hairer E, Lubich C, Wanner G (2002) Geometric Numerical Integration. Structure-Preservings Algorithms for Ordinary Differential Equations. Springer-Verlag Berlin Heidelberg. Vol 31.

41. McLachlan RI, Quispel GRW (2002) Splitting methods. Acta Numer 11: 341–434.

42. McLachlan RI, Quispel GRW (2006) Geometric integrators for ODEs. J Phys A Math Gen 39: 5251–5285.    

43. Forest E (2006) Geometric integration for particle accelerators. J Phys A Math Gen 39: 5351-5377.

44. Blanes S, Casas F, Murua A (2008) Splitting and composition methods in the numerical integration of differential equations. Bol Soc Esp Mat Apl 45: 89–145.

45. Benettin G, Ponno A (2011) On the numerical integration of FPU-like systems. Phys D 240: 568–573.    

46. Antonopoulos Ch, Bountis T, Skokos Ch, et al. (2014) Complex statistics and diffusion in nonlinear disordered particle chains. Chaos 24: 024405.    

47. Antonopoulos Ch, Skokos Ch, Bountis T, et al. (2017) Analyzing chaos in higher order disordered quartic-sextic Klein-Gordon lattices using q-statistics. Chaos Solitons Fractals 104: 129–134.    

48. Tieleman O, Skokos Ch, Lazarides A (2014) Chaoticity without thermalisation in disordered lattices. EPL 105: 20001.    

49. Skokos Ch, Gerlach E, Bodyfelt JD, et al. (2014) High order three part split symplectic integrators: Efficient techniques for the long time simulation of the disordered discrete non linear Schrödinger equation. Phys Lett A 378: 1809–1815.    

50. Gerlach E, Meichsner J, Skokos C (2016) On the symplectic integration of the discrete nonlinear Schrödinger equation with disorder. Eur Phys J Spec Top 225: 1103–1114.    

51. Danieli C, Campbell DK, Flach S (2017) Intermittent many-body dynamics at equilibrium. Phys Rev E 95: 060202.    

52. Thudiyangal M, Kati Y, Danieli C, et al. (2018) Weakly nonergodic dynamics in the Gross-Pitaevskii lattice. Phys Rev Lett 120: 184101.    

53. Thudiyangal M, Danieli C, Kati Y, et al. (2019) Dynamical glass phase and ergodization times in Josephson junction chains. Phys Rev Lett 122: 054102.    

54. Danieli C, Thudiyangal M, Kati Y, et al. (2018) Dynamical glass in weakly non-integrable many-body systems. arXiv:1811.10832.

55. Laskar J, Robutel P (2001) High order symplectic integrators for perturbed Hamiltonian systems. Celest Mech Dyn Astr 80: 39–62.    

56. Senyange B, Skokos Ch (2018) Computational efficiency of symplectic integration schemes: Application to multidimensional disordered Klein-Gordon lattices. Eur Phys J Spec Top 227: 625–643.    

57. Blanes S, Casas F, Farrés A, et al. (2013) New families of symplectic splitting methods for numerical integration in dynamical astronomy. Appl Numer Math 68: 58–72.    

58. Skokos Ch, Gerlach E (2010) Numerical integration of variational equations. Phys Rev E 82: 036704.

59. Gerlach E, Skokos Ch (2011) Comparing the efficiency of numerical techniques for the integration of variational equations: Dynamical systems, differential equations and applications, Discrete & Continuous Dynamical Systems-Supplement 2011, Dedicated to the 8th AIMS Conference, 475–484.

60. Gerlach E, Eggl S, Skokos Ch (2012) Efficient integration of the variational equations of multidimensional Hamiltonian systems: Application to the Fermi-Pasta-Ulam lattice. Int J Bifurcation Chaos 22: 1250216.    

61. Carati A, Ponno A (2018) Chopping time of the FPU α-model. J Stat Phys 170: 883–894.    

62. Flach S, Ivanchenko MV, Kanakov OI (2005) q-Breathers and the Fermi-Pasta-Ulam problem. Phys Rev Lett 95: 064102.    

63. Flach S, Ivanchenko MV, Kanakov OI, et al. (2008) Periodic orbits, localization in normal mode space, and the Fermi-Pasta-Ulam problem. Am J Phys 76: 453.    

64. Flach S, Ponno A (2008) The Fermi-Pasta-Ulam problem: periodic orbits, normal forms and resonance overlap criteria. Phys D 237: 908–917.    

65. Garcia-Mata I, Shepelyansky DL (2009) Delocalization induced by nonlinearity in systems with disorder. Phys Rev E 79: 026205.    

66. Hairer E, Nørsett SP, Wanner G (1993) Solving Ordinary Differential Equations I: Nonstiff Problems, 2Ed., Berlin: Springer, Vol. 14.

67. Barrio R (2005) Performance of the Taylor series method for ODEs/DAEs. Appl Math Comput 163: 525–545.

68. Abad A, Barrio R, Blesa F, et al. (2012) Algorithm 924: TIDES, a Taylor series integrator for differential equations. ACM Math Software 39: 5.

69. Taylor series Integrator for Differential Equations. Available from: https://sourceforge.net/projects/tidesodes/.

70. Gröbner W (1967) Die Lie-Reihen und ihre Anwendungen. Berlin: Deutscher Verlag der Wissenschaften.

71. Blanes S, Casas F (2016) A Concise Introduction to Geometric Numerical Integration. In series of Monographs and Research Notes in Mathematics, Chapman and Hall/CRC.

72. Hanslmeier A, Dvorak R (1984) Numerical integration with Lie series. Astron Astrophys 132: 203–207.

73. Eggl S, Dvorak R (2010) An introduction to common numerical integration codes used in dynamical astronomy. Lect Notes Phys 790: 431–480.    

74. Fortran and Matlab Codes. Available from: http://www.unige.ch/˜hairer/software.html .

75. Boreux J, Carletti T, Hubaux C (2010) High order explicit symplectic integrators for the Discrete Non Linear Schrödinger equation. Report naXys 09, arXiv:1012.3242.

76. Laskar J (2003) Chaos in the solar system. Ann Henri Poincaré 4: 693–705.    

77. Leimkuhler B, Reich S (2004) Simulating Hamiltonian Dynamics. UK: Cambridge University Press, Vol. 14.

78. Lasagni FM (1988) Canonical Runge-Kutta methods. ZAMP 39: 952–953.

79. Sanz-Serna JM (1988) Runge-Kutta schemes for Hamiltonian systems. BIT Numer Math 28: 877–883.    

80. Yoshida H (1990) Construction of higher order symplectic integrator. Phys Lett A 150: 262–268.    

81. Suzuki M (1990) Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys Lett A 146: 319–323.    

82. Yoshida H (1993) Recent progress in the theory and application of symplectic integrators. Celest Mech Dyn Astr 56: 27–43.    

83. Ruth RD (1983) A canonical integration technique. IEEE Trans Nucl Sci 30: 2669–2671.    

84. McLachlan RI (1995) Composition methods in the presence of small parameters. BIT Numer Math 35: 258–268.    

85. Farrés A, Laskar J, Blanes S, et al. (2013) High precision symplectic integrators for the solar system. Celest Mech Dyn Astr 116: 141–174.    

86. Iserles A, Quispel GRW (2018) Why geometric numerical integration? In: Ebrahimi-Fard, K., Barbero Liñán, M. Editors, Discrete Mechanics, Geometric Integration and Lie-Butcher Series, Springer, Cham.

87. Forest E, Ruth RD (1990) Fourth-order symplectic integration. Phys D 43: 105–117.    

88. Kahan W, Li RC (1997) Composition constants for raising the orders of unconventional schemes for ordinary differential equations. Math Comput Am Math Soc 66: 1089–1099.    

89. Sofroniou M, Spaletta G (2005) Derivation of symmetric composition constants for symmetric integrators. Optim Method Soft 20: 597–613.    

90. Blanes S, Moan PC (2001) Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nyström methods. J Comput Appl Math 142: 313–330.

91. Hillebrand M, Kalosakas G, Schwellnus A, et al. (2019) Heterogeneity and chaos in the Peyrard- Bishop-Dauxois DNA model. Phys Rev E 99: 022213.    

92. Laskar J, Vaillant T (2019) Dedicated symplectic integrators for rotation motions. Celest Mech Dyn Astr 131: 15.    

93. Koseleff PV (1996) Exhaustive search of symplectic integrators using computer algebra. Integr Algorithms Classical Mech 10: 103–120.

94. Benettin G, Galgani L, Giorgilli A, et al. (1976) Kolmogorov entropy and numerical experiments. Phys Rev A 14: 2338–2345.    

95. All simulations were performed on the IBS-PCS cluster, which uses Intel(R) Xeon(R) E5-2620 v3 processors. All codes were written in Fortran90 language and were compiled by using the gfortran compiler ( https://gcc.gnu.org/ ) with O3 optimization flag. No advanced vectorization mode has been implemented.

96. All simulations were performed on a workstation using 3.00 GHz Intel Xeon E5-2623 processors. All codes were written in Fortran90 language and were compiled by using the gfortran compiler ( https://gcc.gnu.org/ ) with O3 optimization flag. No advanced vectorization mode has been implemented.

97. Rasmussen K, Cretegny T, Kevrekidis PG, et al. (2000) Statistical mechanics of a discrete nonlinear system. Phys Rev Lett 84: 3740–3743.    

98. Skokos Ch, Manos T (2016) The Smaller (SALI) and the Generalized (GALI) alignment indices: Efficient methods of chaos detection. Lect Notes Phys 915: 129–181.    

99. Achilleos V, Theocharis G, Skokos Ch (2016) Energy transport in one-dimensional disordered granular solids. Phys Rev E 93: 022903.    

100. Livi R, Pettini M, Ruffo S, et al. (1987) Chaotic behavior in nonlinear Hamiltonian systems and equilibrium statistical mechanics. J Stat Phys 48: 539–559.    

101. Binder P, Abraimov D, Ustinov AV, et al. (2000) Observation of breathers in Josephson ladders. Phys Rev Lett 84: 745–748.    

102. Blackburn JA, Cirillo M, Grønbech-Jensen N (2016) A survey of classical and quantum interpretations of experiments on Josephson junctions at very low temperatures. Phys Rep 611: 1–34.    

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