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The European Physical Journal Special Topics

, Volume 227, Issue 5–6, pp 625–643 | Cite as

Computational efficiency of symplectic integration schemes: application to multidimensional disordered Klein–Gordon lattices

  • B. Senyange
  • Ch. Skokos
Regular Article
Part of the following topical collections:
  1. Nonlinear Phenomena in Physics: New Techniques and Applications

Abstract

We implement several symplectic integrators, which are based on two part splitting, for studying the chaotic behavior of one- and two-dimensional disordered Klein–Gordon lattices with many degrees of freedom and investigate their numerical performance. For this purpose, we perform extensive numerical simulations by considering many different initial energy excitations and following the evolution of the created wave packets in the various dynamical regimes exhibited by these models. We compare the efficiency of the considered integrators by checking their ability to correctly reproduce several features of the wave packets propagation, like the characteristics of the created energy distribution and the time evolution of the systems’ maximum Lyapunov exponent estimator. Among the tested integrators the fourth order ABA864 scheme [S. Blanes et al., Appl. Numer. Math. 68, 58 (2013)] showed the best performance as it needed the least CPU time for capturing the correct dynamical behavior of all considered cases when a moderate accuracy in conserving the systems’ total energy value was required. Among the higher order schemes used to achieve a better accuracy in the energy conservation, the sixth order scheme s11ABA82_6 exhibited the best performance.

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References

  1. 1.
    P.W. Anderson, Phys. Rev. 109, 1492 (1958) ADSCrossRefGoogle Scholar
  2. 2.
    D. Shepelyansky, Phys. Rev. Lett. 70, 1787 (1993) ADSCrossRefGoogle Scholar
  3. 3.
    M. Molina, Phys. Rev. B 58, 12547 (1998) ADSCrossRefGoogle Scholar
  4. 4.
    D. Clément, A.F. Varón, M. Hugbart, J.A. Retter, P. Bouyer, L. Sanchez-Palencia, D.M. Gangardt, G.V. Shlyapnikov, A. Aspect, Phys. Rev. Lett. 95, 170409 (2005) ADSCrossRefGoogle Scholar
  5. 5.
    C. Fort, L. Fallani, V. Guarrera, J.E. Lye, M. Modugno, D.S. Wiersma, M. Inguscio, Phys. Rev. Lett. 95 170410 (2005) ADSCrossRefGoogle Scholar
  6. 6.
    T. Schwartz, G. Bartal, S. Fishman, M. Segev, Nature 446, 52 (2007) ADSCrossRefGoogle Scholar
  7. 7.
    Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D.N. Christodoulides, Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008) ADSCrossRefGoogle Scholar
  8. 8.
    J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clément, L. Sanchez-Palencia, P. Bouyer, A. Aspect, Nature 453, 891 (2008) ADSCrossRefGoogle Scholar
  9. 9.
    G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, M. Inguscio, Nature 453, 895 (2008) ADSCrossRefGoogle Scholar
  10. 10.
    G. Kopidakis, S. Komineas, S. Flach, S. Aubry, Phys. Rev. Lett. 100, 084103 (2008) ADSCrossRefGoogle Scholar
  11. 11.
    A.S. Pikovsky, D.L. Shepelyansky, Phys. Rev. Lett. 100, 094101 (2008) ADSCrossRefGoogle Scholar
  12. 12.
    S. Flach, D.O. Krimer, Ch. Skokos, Phys. Rev. Lett. 102, 024101 (2009) ADSCrossRefGoogle Scholar
  13. 13.
    S. Flach, D.O. Krimer, Ch. Skokos, Phys. Rev. Lett. 102, 209903 (2009) ADSCrossRefGoogle Scholar
  14. 14.
    I. García-Mata, D.L. Shepelyansky, Phys. Rev. E 79, 026205 (2009) ADSCrossRefGoogle Scholar
  15. 15.
    Ch. Skokos, D.O. Krimer, S. Komineas, S. Flach, Phys. Rev. E 79, 056211 (2009) ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    H. Veksler, Y. Krivolapov, S. Fishman, Phys. Rev. E 80, 037201 (2009) ADSCrossRefGoogle Scholar
  17. 17.
    Ch. Skokos, S. Flach, Phys. Rev. E 82, 016208 (2010) ADSCrossRefGoogle Scholar
  18. 18.
    T.V. Laptyeva, J.D. Bodyfelt, D.O. Krimer, Ch. Skokos, S. Flach, Europhys. Lett. 91, 30001 (2010) ADSCrossRefGoogle Scholar
  19. 19.
    S. Flach, Chem. Phys. 375, 548 (2010) CrossRefGoogle Scholar
  20. 20.
    D.M. Basko, Ann. Phys. 326, 1577 (2011) ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    M. Mulansky, K. Ahnert, A. Pikovsky, Phys. Rev. E 83, 026205 (2011) ADSCrossRefGoogle Scholar
  22. 22.
    M. Mulansky, K. Ahnert, A. Pikovsky, D.L. Shepelyansky, J. Stat. Phys. 145, 1256 (2011) ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    J.D. Bodyfelt, T.V. Laptyeva, Ch. Skokos, D.O. Krimer, S. Flach, Phys. Rev. E 84, 016205 (2011) ADSCrossRefGoogle Scholar
  24. 24.
    J.D. Bodyfelt, T.V. Laptyeva, G. Gligoric, D.O. Krimer, Ch. Skokos, S. Flach, Int. J. Bifurc. Chaos 21, 2107 (2011) CrossRefGoogle Scholar
  25. 25.
    S.S. Kondov, W.R. McGehee, J.J. Zirbel, B. DeMarco, Science 334, 66 (2011) ADSCrossRefGoogle Scholar
  26. 26.
    T.V. Laptyeva, J.D. Bodyfelt, S. Flach, Europhys. Lett. 98, 60002 (2012) ADSCrossRefGoogle Scholar
  27. 27.
    M. Mulansky, A. Pikovsky, Phys. Rev. E 86, 056214 (2012) ADSCrossRefGoogle Scholar
  28. 28.
    F. Jendrzejewski, A. Bernard, K. Mueller, P. Cheinet, V. Josse, M. Piraud, L. Pezze, L. Sanchez-Palencia, A. Aspect, P. Bouyer, Nat. Phys. 8, 398 (2012) CrossRefGoogle Scholar
  29. 29.
    M. Mulansky, A. Pikovsky, New J. Phys. 15, 053015 (2013) ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Ch. Skokos, I. Gkolias, S. Flach, Phys. Rev. Lett. 111, 064101 (2013) ADSCrossRefGoogle Scholar
  31. 31.
    Ch. Antonopoulos, T. Bountis, Ch. Skokos, L. Drossos, Chaos 24, 024405 (2014) ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    T.V. Laptyeva, M.V. Ivanchenko, S. Flach, J. Phys. A 47, 493001 (2014) Google Scholar
  33. 33.
    Ch. Skokos, D.O. Krimer, S. Komineas, S. Flach, Phys. Rev. E 89, 029907 (2014) ADSCrossRefGoogle Scholar
  34. 34.
    O. Tieleman, Ch. Skokos, A. Lazarides, Europhys. Lett. 105, 20001 (2014) ADSCrossRefGoogle Scholar
  35. 35.
    A.J. Martínez, P.G. Kevrekidis, M.A. Porter, Phys. Rev. E 93, 022902 (2016) ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    V. Achilleos, G. Theocharis, Ch. Skokos, Phys. Rev. E 93, 022903 (2016) ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    A.J. Martínez, H. Yasuda, E. Kim, P.G. Kevrekidis, M.A. Porter, J. Yang, Phys. Rev. E 93, 052224 (2016) ADSCrossRefGoogle Scholar
  38. 38.
    Ch. Antonopoulos, Ch. Skokos, T. Bountis, S. Flach, Chaos, Solitons Fractals 104, 129 (2017) ADSCrossRefGoogle Scholar
  39. 39.
    C. Chong, M.A. Porter, P.G. Kevrekidis, C. Daraio, J. Phys. Condens. Matter 29, 413003 (2017) CrossRefGoogle Scholar
  40. 40.
    S. Donsa, H. Hofstätter, O. Koch, J. Burgdörfer, I. Březinová, Phys. Rev. A 96, 043630 (2016) CrossRefGoogle Scholar
  41. 41.
    V. Achilleos, G. Theocharis, Ch. Skokos, Phys. Rev. E 97, 042220 (2018) ADSCrossRefGoogle Scholar
  42. 42.
    E. Hairer, C. Lubich, G. Wanner, Structure-preserving algorithms for ordinary differential equations in Geometric Numerical Integration, Springer Series in Computational Mathematics (Springer, New York, 2002), Vol. 31, Chap. VI Google Scholar
  43. 43.
    R.I. McLachan, G.R.W. Quispel, Acta Numer. 11, 341 (2002) MathSciNetGoogle Scholar
  44. 44.
    R.I. McLachan, G.R.W. Quispel, J. Phys. A 39, 5251 (2006) ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    E. Forest, J. Phys. A 39, 5321 (2006) ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    S. Blanes, F. Casas, A. Murua, Bol. Soc. Esp. Mat. Apl. 45, 89 (2008) Google Scholar
  47. 47.
    Ch. Skokos, E. Gerlach, J.D. Bodyfelt, G. Papamikos, S. Eggl, Phys. Lett. A 378, 1809 (2014) ADSMathSciNetCrossRefGoogle Scholar
  48. 48.
    E. Gerlach, J. Meichsner, Ch. Skokos, Eur. Phys. J. Spec. Topics 225, 1103 (2016) ADSCrossRefGoogle Scholar
  49. 49.
    J. Laskar, P. Robutel, Cel. Mech. Dyn. Astron. 80, 39 (2001) ADSCrossRefGoogle Scholar
  50. 50.
    Ch. Skokos, G.A. Gottwald, J. Laskar, in Chaos Detection and Predictability, Lecture Notes in Physics (Springer Verlag, Berlin, 2016), Vol. 915 Google Scholar
  51. 51.
    G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Meccanica 15, 9 (1980) ADSCrossRefGoogle Scholar
  52. 52.
    G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Meccanica 15, 21 (1980) CrossRefGoogle Scholar
  53. 53.
    Ch. Skokos, Lect. Notes Phys. 790, 63 (2010) ADSCrossRefGoogle Scholar
  54. 54.
    R.D. Ruth, IEEE Trans. Nucl. Sci. 30, 2669 (1983) ADSCrossRefGoogle Scholar
  55. 55.
    R.I. Mclachlan, BIT 35, 258 (1995) MathSciNetCrossRefGoogle Scholar
  56. 56.
    A. Farres, J. Laskar, S. Blanes, F. Casas, J. Makazaga, A. Murua, Cel. Mech. Dyn. Astron. 116, 141 (2013) ADSCrossRefGoogle Scholar
  57. 57.
    H. Yoshida, Phys. Lett. A 150, 262 (1990) ADSMathSciNetCrossRefGoogle Scholar
  58. 58.
    E. Forest, R.D. Ruth, Phys. D 43, 105 (1990) MathSciNetCrossRefGoogle Scholar
  59. 59.
    S. Blanes, F. Casas, A. Farres, J. Laskar, J. Makazaga, A. Murua, Appl. Numer. Math. 68, 58 (2013) MathSciNetCrossRefGoogle Scholar
  60. 60.
    W. Kahan, R. Li, Math. Comput. 66, 1089 (1997) ADSCrossRefGoogle Scholar
  61. 61.
    M. Sofroniou, G. Spaletta, Opt. Meth. Soft. 20, 597 (2005) CrossRefGoogle Scholar
  62. 62.
    G. Benettin, L. Galgani, J.-M. Strelcyn, Phys. Rev. A 14, 2338 (1976) ADSCrossRefGoogle Scholar
  63. 63.
    Ch. Skokos, E. Gerlach E., Phy. Rev. E 82, 036704 (2010) ADSCrossRefGoogle Scholar
  64. 64.
    E. Gerlach, Ch. Skokos, Discr. Contin. Dyn. Syst. Supp. 2011, 475 (2011) Google Scholar
  65. 65.
    E. Gerlach, S. Eggl, Ch. Skokos, Int. J. Bifurc. Chaos 22, 1250216 (2012) CrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of Cape TownCape TownSouth Africa

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