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Heterogeneous multiscale methods for stiff ordinary differential equations


Authors: Bjorn Engquist and Yen-Hsi Tsai
Journal: Math. Comp. 74 (2005), 1707-1742
MSC (2000): Primary 65Lxx, 65Pxx, 37Mxx
DOI: https://doi.org/10.1090/S0025-5718-05-01745-X
Published electronically: May 18, 2005
MathSciNet review: 2164093
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Abstract: The heterogeneous multiscale methods (HMM) is a general framework for the numerical approximation of multiscale problems. It is here developed for ordinary differential equations containing different time scales. Stability and convergence results for the proposed HMM methods are presented together with numerical tests. The analysis covers some existing methods and the new algorithms that are based on higher-order estimates of the effective force by kernels satisfying certain moment conditions and regularity properties. These new methods have superior computational complexity compared to traditional methods for stiff problems with oscillatory solutions.


References [Enhancements On Off] (What's this?)

  • 1. V. I. Arnol'd.
    Mathematical methods of classical mechanics.
    Springer-Verlag, New York, 1989.
    Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition. MR 1345386 (96c:70001)
  • 2. N. N. Bogoliubov and Y. A. Mitropolsky.
    Asymptotic methods in the theory of non-linear oscillations.
    Translated from the second revised Russian edition. International Monographs on Advanced Mathematics and Physics. Hindustan Publishing Corp., Delhi, Gordon and Breach Science Publishers, New York, 1961. MR 0141845 (25:5242)
  • 3. G. Browning and H.-O. Kreiss.
    Splitting methods for problems with different timescales.
    Monthly Weather Review, 122(11):2614-2622, 1994.
  • 4. Germund G. Dahlquist, Werner Liniger, and Olavi Nevanlinna.
    Stability of two-step methods for variable integration steps.
    SIAM J. Numer. Anal., 20(5):1071-1085, 1983. MR 0714701 (85b:65079)
  • 5. Weinan E.
    Analysis of the heterogeneous multiscale method for ordinary differential equations.
    Commun. Math. Sci., 1(3):423-436, 2003. MR 2069938
  • 6. Weinan E and Bjorn Engquist.
    The heterogeneous multiscale methods.
    Commun. Math. Sci., 1(1):87-132, 2003. MR 1979846 (2004b:35019)
  • 7. Weinan E, Di Liu, and Eric Vanden-Eijnden.
    Analysis of multiscale techniques for stochastic dynamical systems.
  • 8. Weinan E and Eric Vanden-Eijnden.
    Numerical techniques for multiscale dynamical systems with stochastic effects.
    Comm. Math. Sci., 1(2):385-391, 2003. MR 1980483
  • 9. B. García-Archilla, J. M. Sanz-Serna, and R. D. Skeel.
    Long-time-step methods for oscillatory differential equations.
    SIAM J. Sci. Comput., 20(3):930-963 (electronic), 1999. MR 1648882 (99g:65087)
  • 10. C. W. Gear and D. R. Wells.
    Multirate linear multistep methods.
    BIT, 24(4):484-502, 1984. MR 0764821 (86a:65057)
  • 11. C.W. Gear and Iaonnis G. Kevrekidis.
    ``Coarse" integration/bifurcation analysis via microscopic simulators: micro-Galerkin methods.
    2002.
  • 12. O. F. Graf and D. G. Bettis.
    Modified multirevolution integration methods for satellite orbit computation.
    Celestial Mech., 11:433-448, 1975. MR 0373409 (51:9609)
  • 13. E. Hairer and G. Wanner.
    Solving ordinary differential equations. II, volume 14 of Springer Series in Computational Mathematics.
    Springer-Verlag, Berlin, second edition, 1996.
    Stiff and differential-algebraic problems. MR 1439506 (97m:65007)
  • 14. Ernst Hairer, Christian Lubich, and Gerhard Wanner.
    Geometric numerical integration, volume 31 of Springer Series in Computational Mathematics.
    Springer-Verlag, Berlin, 2002.
    Structure-preserving algorithms for ordinary differential equations. MR 1904823 (2003f:65203)
  • 15. Arieh Iserles.
    On the global error of discretization methods for highly-oscillatory ordinary differential equations.
    BIT, 42(3):561-599, 2002. MR 1931887 (2003i:65056)
  • 16. Tosio Kato.
    Perturbation theory for linear operators.
    Classics in Mathematics. Springer-Verlag, Berlin, 1995.
    Reprint of the 1980 edition. MR 1335452 (96a:47025)
  • 17. N. Kopell.
    Invariant manifolds and the initialization problem for some atmospheric equations.
    Phys. D, 14(2):203-215, 1985. MR 0784954 (86g:86005)
  • 18. Heinz-Otto Kreiss.
    Difference methods for stiff ordinary differential equations II.
    Preprint.
  • 19. Heinz-Otto Kreiss.
    Problems with different time scales for ordinary differential equations.
    SIAM J. Numer. Anal., 16(6):980-998, 1979. MR 0551320 (81a:65087)
  • 20. Heinz-Otto Kreiss and Jens Lorenz.
    Manifolds of slow solutions for highly oscillatory problems.
    Indiana Univ. Math. J., 42(4):1169-1191, 1993. MR 1266089 (95f:34077)
  • 21. V. I. Lebedev and S. A. Finogenov.
    The use of ordered Cebysev parameters in iteration methods.
    Z. Vycisl. Mat. i Mat. Fiz., 16(4):895-907, 1084, 1976. MR 0443314 (56:1684)
  • 22. Ben Leimkuhler and Sebastian Reich.
    A reversible averaging integrator for multiple time-scale dynamics.
    J. Comput. Phys., 171(1):95-114, 2001. MR 1843642 (2002d:65064)
  • 23. Di Liu.
    Topics in the analysis and computation of stochastic differential equations.
    Ph.D. thesis, Princeton University, 2003.
  • 24. David Mace and L. H. Thomas.
    An extrapolation formula for stepping the calculation of the orbit of an artificial satellite several revolutions ahead at a time.
    Astronom. J., 65:300-303, 1960. MR 0119998 (22:10755)
  • 25. George Majda.
    Filtering techniques for systems of stiff ordinary differential equations. I.
    SIAM J. Numer. Anal., 21(3):535-566, 1984. MR 0744172 (85i:65089)
  • 26. George Majda.
    Filtering techniques for systems of stiff ordinary differential equations. II. Error estimates.
    SIAM J. Numer. Anal., 22(6):1116-1134, 1985. MR 0811187 (87d:65074)
  • 27. W. L. Miranker and G. Wahba.
    An averaging method for the stiff highly oscillatory problem.
    Math. Comp., 30(135):383-399, 1976. MR 0423817 (54:11791)
  • 28. Linda R. Petzold.
    An efficient numerical method for highly oscillatory ordinary differential equations.
    SIAM J. Numer. Anal., 18(3):455-479, 1981. MR 0615526 (82h:65059)
  • 29. Linda R. Petzold, Laurent O. Jay, and Jeng Yen.
    Numerical solution of highly oscillatory ordinary differential equations.
    In Acta numerica, 1997, volume 6 of Acta Numer., pages 437-483. Cambridge Univ. Press, Cambridge, 1997. MR 1489260 (98k:65040)
  • 30. Robert E. Scheid, Jr.
    The accurate numerical solution of highly oscillatory ordinary differential equations.
    Math. Comp., 41(164):487-509, 1983. MR 0717698 (85i:65091)
  • 31. Robert E. Scheid, Jr.
    Difference methods for problems with different time scales.
    Math. Comp., 44(169):81-92, 1985. MR 0771032 (86e:65098)
  • 32. B. Engquist, P. Lötstedt, and O. Runborg (Eds.).
    Lecture Notes in Computational Science and Engineering, Vol. 44, Springer, 2005.
  • 33. Michael Alan Zeitzew.
    Numerical Methods for Nonlinear Ordinary Differential Equations with Different Time-Scales.
    Ph.D. thesis, UCLA, 1995.
  • 34. Abraham Ziv and Vardy Amdursky.
    On the numerical solution of stiff linear systems of the oscillatory type.
    SIAM J. Appl. Math., 33(4):593-606, 1977. MR 0455417 (56:13655)

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Additional Information

Bjorn Engquist
Affiliation: Department of Mathematics and PACM, Princeton University, Princeton, New Jersey 08544, and Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden

Yen-Hsi Tsai
Affiliation: Institute for Advanced Study and Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: ytsai@math.princeton.edu

DOI: https://doi.org/10.1090/S0025-5718-05-01745-X
Received by editor(s): July 31, 2003
Received by editor(s) in revised form: June 4, 2004
Published electronically: May 18, 2005
Additional Notes: The second author is partially supported by the National Science Foundation under agreement No. DMS-0111298
Article copyright: © Copyright 2005 American Mathematical Society

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