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Applied Mathematics and Mechanics

, Volume 39, Issue 11, pp 1529–1546 | Cite as

Clean numerical simulation: a new strategy to obtain reliable solutions of chaotic dynamic systems

  • Xiaoming Li
  • Shijun Liao
Article
  • 11 Downloads

Abstract

It is well-known that chaotic dynamic systems, e.g., three-body system and turbulent flow, have sensitive dependence on the initial conditions (SDIC). Unfortunately, numerical noises, i.e., truncation error and round-off error, always exist in practice. Thus, due to the SDIC, the long-term accurate prediction of chaotic dynamic systems is practically impossible. In this paper, a new strategy for chaotic dynamic systems, i.e., the clean numerical simulation (CNS), is briefly described, and applied to a few Hamiltonian chaotic systems. With negligible numerical noises, the CNS can provide convergent (reliable) chaotic trajectories in a long enough interval of time. This is very important for Hamiltonian systems, and thus should have many applications in various fields. It is found that the traditional numerical methods in double precision cannot give not only reliable trajectories but also reliable Fourier power spectra and autocorrelation functions (ACFs). In addition, even the statistic properties of chaotic systems cannot be correctly obtained by means of traditional numerical algorithms in double precision, as long as these statistics are time-dependent. The CNS results strongly suggest that one had better to be very careful on the direct numerical simulation (DNS) results of statistically unsteady turbulent flows, although DNS results often agree well with experimental data when the turbulent flow is in a statistical stationary state.

Key words

chaos numerical noise clean numerical simulation (CNS) reliability of computation 

Chinese Library Classification

O415.6 

2010 Mathematics Subject Classification

37M05 65P20 

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Notes

Acknowledgement

We thank the anonymous reviewers for their valuable comments and suggestions.

References

  1. [1]
    POINCARÉ, J. H. Sur le problème des trois corps et les équations de la dynamique. Divergence des séries de M. Lindstedt. Acta Mathematica, 13, 1–270 (1890)zbMATHGoogle Scholar
  2. [2]
    LORENZ, E. N. Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20, 130–141 (1963)CrossRefzbMATHGoogle Scholar
  3. [3]
    LORENZ, E. N. Computational chaos-a prelude to computational instability. Physica D, 15, 299–317 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    LORENZ, E. N. Computational periodicity as observed in a simple system. Tellus A, 58, 549–559 (2006)CrossRefGoogle Scholar
  5. [5]
    LI, J. P., ZENG, Q. C., and CHOU, J. F. Computational uncertainty principle in nonlinear ordinary differential equations (II): theoretical analysis. Science in China (Series E), 44, 55–74 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    TEIXEIRA, J., REYNOLDS, C., and JUDD, K. Time step sensitivity of nonlinear atmospheric models: numerical convergence, truncation error growth, and ensemble design. Journal of the Atmospheric Sciences, 64, 175–188 (2007)CrossRefGoogle Scholar
  7. [7]
    QIN, S. J. and LIAO, S. J. Influence of round-off errors on the reliability of numerical simulations of chaotic dynamic systems. Journal of Applied Nonlinear Dynamics (accepted) (Preprint arXiv:1707.04720)Google Scholar
  8. [8]
    YAO, L. and HUGHES, D. Comment on “computational periodicity as observed in a simple system” by Edward N. Lorenz (2006). Tellus A, 60, 803–805 (2008)CrossRefGoogle Scholar
  9. [9]
    LORENZ, E. N. Reply to comment by L. S. Yao and D. Hughes. Tellus A, 60, 806–807 (2008)CrossRefGoogle Scholar
  10. [10]
    ALBERS, T. and RADONS, G. Weak ergodicity breaking and aging of chaotic transport in Hamiltonian systems. Physical Review Letters, 113, 184101 (2014)CrossRefGoogle Scholar
  11. [11]
    HUYNH, H. N., NGUYEN, T. P. T., and CHEW, L. Y. Numerical simulation and geometrical analysis on the onset of chaos in a system of two coupled pendulums. Communications in Nonlinear Science and Numerical Simulation, 18, 291–307 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    LEE, M. and MOSER, R. D. Direct numerical simulation of turbulent channel flow up to to Re τ ≈ 5200. Journal of Fluid Mechanics, 774, 395–415 (2015)CrossRefGoogle Scholar
  13. [13]
    WANG, J. C., LI, Q. X., and E, W.N. Study of the instability of the Poiseuille flow using a thermodynamic formalism. Proceedings of the National Academy of Sciences, 112, 9518–9523 (2015)CrossRefGoogle Scholar
  14. [14]
    AVILA, K., MOXEY, D., de LOZAR, A., AVILA, M., BARKLEY, D., and HOF, B. The onset of turbulence in pipe flow. Science, 333, 192–196 (2011)CrossRefGoogle Scholar
  15. [15]
    DEIKE, L., FUSTER, D., BERHANU, M., and FALCON, E. Direct numerical simulations of capillary wave turbulence. Physical Review Letters, 112, 234501 (2014)CrossRefGoogle Scholar
  16. [16]
    KIM, J., MOIN, P., and MOSER, R. Turbulence statistics in fully developed channel flow at low Reynolds number. Journal of Fluid Mechanics, 177, 133–166 (1987)CrossRefzbMATHGoogle Scholar
  17. [17]
    YEE, H., TORCZYNSKI, J., MORTON, S., VISBAL, M., and SWEBY, P. On spurious behavior of CFD simulations. International Journal for Numerical Methods in Fluids, 30, 675–711 (1999)CrossRefzbMATHGoogle Scholar
  18. [18]
    WANG, L. P. and ROSA, B. A spurious evolution of turbulence originated from round-off error in pseudo-spectral simulation. Computers and Fluids, 38, 1943–1949 (2009)CrossRefzbMATHGoogle Scholar
  19. [19]
    YEE, H. C., SWEBY, P. K., and GRIFFITHS, D. F. Dynamical approach study of spurious steadystate numerical solutions of nonlinear differential equations, I: the dynamics of time discretization and its implications for algorithm development in computational fluid dynamics. Journal of Computational Physics, 97, 249–310 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    YEE, H. C. and SWEBY, P. K. Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations, II: Global asymptotic behavior of time discretizations. International Journal of Computational Fluid Dynamics, 4, 219–283 (1995)CrossRefGoogle Scholar
  21. [21]
    KRYS’KO, V. A., AWREJCEWICZ, J., and BRUK, V. M. On the solution of a coupled thermomechanical problem for non-homogeneous Timoshenko-type shells. Journal of Mathematical Analysis and Applications, 273, 409–416 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    AWREJCEWICZ, J. and KRYSKO, V. A. Nonlinear coupled problems in dynamics of shells. International Journal of Engineering Science, 41, 587–607 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    AWREJCEWICZ, J., KRYSKO, V. A., and KRYSKO, A. V. Complex parametric vibrations of flexible rectangular plates. Meccanica, 39, 221–244 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    AWREJCEWICZ, J., KRYSKO, A. V., ZHIGALOV, M. V., SALTYKOVA, O. A., and KRYSKO, V. A. Chaotic vibrations in flexible multi-layered Bernoulli-Euler and Timoshenko type beams. Latin American Journal of Solids and Structures, 5, 319–363 (2008)Google Scholar
  25. [25]
    AWREJCEWICZ, J., KRYSKO, A. V., KUTEPOV, I. E., ZAGNIBORODA, N. A., DOBRIYAN, V., and KRYSKO, V. A. Chaotic dynamics of flexible Euler-Bernoulli beams. Chaos, 34, 043130 (2014)MathSciNetzbMATHGoogle Scholar
  26. [26]
    KRYSKO, A. V., AWREJCEWICZ, J., SALTYKOVA, O. A., ZHIGALOV, M. V., and KRYSKO, V. A. Investigations of chaotic dynamics of multi-layer beams using taking into account rotational inertial effects. Communications in Nonlinear Science and Numerical Simulation, 19, 2568–2589 (2014)MathSciNetCrossRefGoogle Scholar
  27. [27]
    AWREJCEWICZ, J., KRYSKO, V. A. J., PAPKOVA, I. V., KRYLOV, E. Y., and KRYSKO, A. V. Spatio-temporal non-linear dynamics and chaos in plates and shells. Nonlinear Studies, 21, 313–327 (2004)MathSciNetzbMATHGoogle Scholar
  28. [28]
    AWREJCEWICZ, J., KRYSKO, A. V., ZAGNIBORODA, N. A., DOBRIYAN, V. V., and KRYSKO, V. A. On the general theory of chaotic dynamic of flexible curvilinear Euler-Bernoulli beams. Nonlinear Dynamics, 85, 2729–2748 (2016)CrossRefzbMATHGoogle Scholar
  29. [29]
    AWREJCEWICZ, J., KRYSKO, A. V., PAPKOVA, I. V., ZAKHAROV, V. M., EROFEEV, N. P., KRYLOVA, E. Y., MROZOWSKI, J., and KRYSKO, V. A. Chaotic dynamics of flexible beams driven by external white noise. Mechanical Systems and Signal Processing, 79, 225–253 (2016)CrossRefGoogle Scholar
  30. [30]
    AWREJCEWICZ, J., KRYSKO, A. V., EROFEEV, N. P., DOBRIYAN, V., BARULINA, M. A., and KRYSKO, V. A. Quantifying chaos by various computational methods, part 1: simple systems. Entropy, 20, 175 (2018)MathSciNetCrossRefGoogle Scholar
  31. [31]
    AWREJCEWICZ, J., KRYSKO, A. V., EROFEEV, N. P., DOBRIYAN, V., BARULINA, M. A., and KRYSKO, V. A. Quantifying chaos by various computational methods, part 2: vibrations of the Bernoulli-Euler beam subjected to periodic and colored noise. Entropy, 20, 170 (2018)CrossRefGoogle Scholar
  32. [32]
    LIAO, S. J. On the reliability of computed chaotic solutions of non-linear differential equations. Tellus A, 61, 550–564 (2009)CrossRefGoogle Scholar
  33. [33]
    WANG, P. F., LI, J. P., and LI, Q. Computational uncertainty and the application of a high-performance multiple precision scheme to obtaining the correct reference solution of Lorenz equations. Numerical Algorithms, 59, 147–159 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    LIAO, S. J. Physical limit of prediction for chaotic motion of three-body problem. Communications in Nonlinear Science and Numerical Simulation, 19, 601–616 (2014)MathSciNetCrossRefGoogle Scholar
  35. [35]
    LIAO, S. J. and WANG, P. F. On the mathematically reliable long-term simulation of chaotic solutions of Lorenz equation in the interval [0, 10 000]. Science China: Physics, Mechanics and Astronomy, 57, 330–335 (2014)CrossRefGoogle Scholar
  36. [36]
    LIAO, S. J. Can we obtain a reliable convergent chaotic solution in any given finite interval of time? International Journal of Bifurcation and Chaos, 24, 1450119 (2014)CrossRefzbMATHGoogle Scholar
  37. [37]
    LI, X. M. and LIAO, S. J. On the stability of the three classes of Newtonian three-body planar periodic orbits. Science China: Physics, Mechanics and Astronomy, 57, 2121–2126 (2014)Google Scholar
  38. [38]
    LIAO, S. J. and LI, X. M. On the inherent self-excited macroscopic randomness of chaotic three-body systems. International Journal of Bifurcation and Chaos, 25, 1530023 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    LIN, Z. L., WANG, L. P., and LIAO, S. J. On the origin of intrinsic randomness of Rayleigh-Bénard turbulence. Science China: Physics, Mechanics and Astronomy, 60, 014712 (2017)Google Scholar
  40. [40]
    LI, X. M. and LIAO, S. J. More than six hundred new families of Newtonian periodic planar collisionless three-body orbits. Science China: Physics, Mechanics and Astronomy, 60, 129511 (2017)Google Scholar
  41. [41]
    LIAO, S. J. On the clean numerical simulation (CNS) of chaotic dynamic systems. Journal of Hydrodynamics, 29, 729–747 (2017)CrossRefGoogle Scholar
  42. [42]
    LI, X. M., JING, Y. P., and LIAO, S. J. Over a thousand new periodic orbits of planar three-body system with unequal mass. Publications of the Astronomical Society of Japan, 70, 64 (2018)Google Scholar
  43. [43]
    BARTON, D., WILLERS, I. M., and ZAHAR, R. V. M. The automatic solution of systems of ordinary differential equations by the method of Taylor series. The Computer Journal, 14, 243–248 (1971)CrossRefzbMATHGoogle Scholar
  44. [44]
    CORLISS, G. and LOWERY, D. Choosing a stepsize for Taylor series methods for solving ODEs. Journal of Computational and Applied Mathematics, 3, 251–256 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    CORLISS, G. and CHANG, Y. F. Solving ordinary differential equations using Taylor series. ACM Transactions on Mathematical Software, 8, 114–144 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    JORBA, A. and ZOU, M. R. A software package for the numerical integration of ODEs by means of high-order Taylor methods. Experimental Mathematics, 14, 99–117 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    BARRIO, R., BLESA, F., and LARA, M. VSVO formulation of the Taylor method for the numerical solution of ODEs. Computers and Mathematics with Applications, 50, 93–111 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    PORTILHO, O. MP—a multiple precision package. Computer Physics Communications, 59, 345–358 (1990)CrossRefGoogle Scholar
  49. [49]
    SUN, B. Kepler’s third law of n-body periodic orbits in a Newtonian gravitation field. Science China: Physics, Mechanics and Astronomy, 61, 054721 (2018)Google Scholar
  50. [50]
    FRISCH, A., MARK, M., AIKAWA, K., FERLAINO, F., BOHN, J. L., MAKRIDES, C., PETROV, A., and KOTOCHIGOVA, S. Quantum chaos in ultracold collisions of gas-phase erbium atoms. nature, 507, 475–479 (2014)CrossRefGoogle Scholar
  51. [51]
    SUSSMAN, G. J. and WISDOM, J. Chaotic evolution of the solar system. Science, 257, 56–62 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    MCLACHLAN, R. I., MODIN, K., and VERDIER, O. Symplectic integrators for spin systems. Physical Review E, 89, 061301 (2014)CrossRefzbMATHGoogle Scholar
  53. [53]
    LASKAR, J. and ROBUTEL, P. High order symplectic integrators for perturbed Hamiltonian systems. Celestial Mechanics and Dynamical Astronomy, 80, 39–62 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    QIN, H. and GUAN, X. Variational symplectic integrator for long-time simulations of the guiding-center motion of charged particles in general magnetic fields. Physical Review Letters, 100, 035006 (2008)CrossRefGoogle Scholar
  55. [55]
    FARRÉS, A., LASKAR, J., BLANES, S., CASAS, F., MAKAZAGA, J., and MURUA, A. High precision symplectic integrators for the solar system. Celestial Mechanics and Dynamical Astronomy, 116, 141–174 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    FOREST, E. and RUTH, R. D. Fourth-order symplectic integration. Physica D: Nonlinear Phenomena, 43, 105–117 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    YOSHIDA, H. Construction of higher order symplectic integrators. Physics Letters A, 150, 262–268 (1990)MathSciNetCrossRefGoogle Scholar
  58. [58]
    HÉNON, M. and HEILES, C. The applicability of the third integral of motion: some numerical experiments. The Astronomical Journal, 69, 73–79 (1964)MathSciNetCrossRefGoogle Scholar
  59. [59]
    SPROTT, J. C. Elegant Chaos: Algebraically Simple Chaotic Flows, World Scientific, Singapore (2010)CrossRefzbMATHGoogle Scholar
  60. [60]
    LIAO, S. J. On the numerical simulation of propagation of micro-level inherent uncertainty for chaotic dynamic systems. Chaos, Solitons & Fractals, 47, 1–12 (2013)MathSciNetCrossRefGoogle Scholar
  61. [61]
    SALTZMAN, B. Finite amplitude free convection as an initial value problem-I. Journal of the Atmospheric Sciences, 19, 329–341 (1962)CrossRefGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Naval Architecture, Ocean and Civil EngineeringShanghai Jiao Tong UniversityShanghaiChina
  2. 2.State Key Laboratory of Ocean EngineeringShanghaiChina
  3. 3.Collaborative Innovative Center for Advanced Ship and Deep-Sea ExplorationShanghaiChina
  4. 4.Ministry-of-Education Key Laboratory of Scientific and Engineering ComputingShanghaiChina

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