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On the reliability of computed chaotic solutions of non-linear differential equations Cover

On the reliability of computed chaotic solutions of non-linear differential equations

By:   
Open Access
|Jan 2009

References

  1. Ashwin , P. 2003 . Synchronization from chaos . Nature 42 , 384 385 .
  2. Cloutman , L. D. 1996 . A note on the stability and accuracy of finite differ-ence approximations to differential equations. Report No. UCRL-ID-125549, Lawrence Livermore National Laboratory, Livermore, CA .
  3. Cloutman , L. D. 1998 . Chaos and instabilities in finite difference approx-imations to nonlinear differential equations. Report No. UCRL-ID-131333, Lawrence Livermore National Laboratory, Livermore, CA .
  4. Einstein , E. 1905 . On the electrodynamics of moving bodies . Annalen der Physik 17 , 891 921 .
  5. Hanski , I. , Turchin , P. , Korplmäki , E. and Henttonen , H. 1993 . Popula-tion oscillations of boreal rodents: regulation by mustelid predators leads to chaos . Nature 364 , 232 235 .
  6. Heisenberg , W. 1927 . -Ober den anschaulichen Inhalt der quantentheo-retischen Kinematik und Mechanilc . Zeitschrift fiir Physik 43 , 172 198 .
  7. Le , H. , Moin , P. and Kim , J. 1997 . Direct numerical simulation of turbulent flow over a backward-facing step , J. Fluid Mech . 330 , 349 374 .
  8. Li , T. Y. and Yorke , J. A. 1975 . Period three implies chaos . Am. Math. Mon . 10 , 985 992 .
  9. Lorenz , E. N. 1963 . Deterministic nonperiodic flow. J. Atmos. Sc i . 20 , 130 141 .
  10. Lorenz , E. N. 1989 . Computational chaos: a prelude to computational instability . Physica D 35 , 299 317 .
  11. Lorenz , E. N. 2006 . Computational periodicity as observed in a simple system . Tellus 58A , 549 557 .
  12. Malescio , G. 2005 . Predicting with unpredictability . Nature 434 , 1073 .
  13. Martin , M. P. , Taylor , E. M. , Wu , M. and Weirs , V. G. 2006 . A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence . J. Comp. Phys . 220 , 270 289 .
  14. Moin , P. and Mahesh , K. 1998 . Direct numerical simulation: a tool in turbulence research . Ann. Rev. Fluid Mech . 30 , 539 578 .
  15. Moser , R. D. , Kim , J. and Mansour , N. N. 1999 . Direct numerical simulation of turbulent channel flow up to Re = 590 . Phys. Fluids 11 , 943 945 .
  16. Parker , T. S. and Chua , L. 0. 1989 . Practical Numerical Algorithms for Chaotic Systems . Springer-Verlag , New York .
  17. Scardovelli , R. and Zaleski , S. 1999 . Direct numerical simulation of free-surface and interfacial flow . Ann. Rev. Fluid Mech . 31 , 567 603 .
  18. Stewart , I. 2000 . The Lorenz attractor exists . Nature 406 , 948 949 .
  19. Teixeira , J. , Reynolds , C. A. and Judd , K. 2007 . Time step sensitivity of nonlinear atmospheric models: numerical convergence, truncation error growth, and ensemble design. J. Atmos. Sc i . 64 , 175 189 .
  20. Teixeira , J. , Reynolds , C. A. and Judd , K. 2008 . Reply to Yao and Hughes’ comments. J. Atmos. Sc i . 65 , 683 684 .
  21. Yao , L. S. and Hughes , D. 2008a . Comment on ‘Computational period-icity as observed in a simple system’ By Edward N. Lorenz (2006) . Tellus 60A , 803 805 .
  22. Yao , L. S. and Hughes , D. 2008b . Comments on ‘Time step sensitivity of nonlinear atmospheric models: numerical convergence, truncation error growth, and ensemble design’. J. Atmos. Sc i . 65 , 681 682 .
Language: English
Page range: 550 - 564
Submitted on: Oct 14, 2008
Accepted on: Mar 5, 2009
Published on: Jan 1, 2009
Published by: Stockholm University Press
In partnership with: Paradigm Publishing Services

© 2009 Shijun Liao, published by Stockholm University Press
This work is licensed under the Creative Commons Attribution 4.0 License.