Mathematics of Computation

A multiscale method for highly oscillatory ordinary differential equations with resonance

Author(s): Gil Ariel; Bjorn Engquist; Richard Tsai.
Journal: Math. Comp. 78 (2009), 929-956.
MSC (2000): Primary 65L05, 34E13, 34E20
Posted: October 3, 2008
Retrieve article in: PDF

Abstract: A multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations (ODEs) is presented. The oscillations may be in resonance with one another and thereby generate hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables whose dynamics is closed up to $\epsilon$ perturbation, and is sufficient to approximate any variable and functional that are slow under the dynamics of the ODE. This set of variables is detected numerically as a preprocessing step in the numerical methods. Error and complexity estimates are obtained. The advantages of the method is demonstrated with a few examples, including a commonly studied problem of Fermi, Pasta, and Ulam.

1.: A. Abdulle.
Fourth order Chebyshev methods with recurrence relation.
SIAM J. Sci. Comput., 23(6):2041-2054 (electronic), 2002. MR 1923724 (2003g:65074)
2.: G. Ariel, B. Engquist, and R. Tsai.
A multiscale method for weakly coupled oscillators.
Submitted to Multi. Model. Simul.
3.: N. N. Bogoliubov and Yu. A. Mitropolski.
Asymptotic Methods in the Theory of Nonlinear Oscillations.
Gordon and Breach, New York, 1961.
4.: G. L. Browning and H.-O. Kreiss.
Multiscale bounded derivative initialization for an arbitrary domain.
J. Atmospheric Sci., 59(10):1680-1696, 2002. MR 1905843 (2003c:86003)
5.: G. Dahlquist.
Convergence and stability in the numerical integration of ordinary differential equations.
Mathematica Scandinavica, 4:33-53, 1956. MR 0080998 (18:338d)
6.: G. Dahlquist.
Stability and error bounds in the numerical integration of ordinary differential equations.
Kungl. Tekn. Högsk. Handl. Stockholm. No., 130:87, 1959. MR 0102921 (21:1706)
7.: G. Dahlquist.
A special stability problem for linear multistep methods.
Nordisk Tidskr. Informations-Behandling, 3:27-43, 1963. MR 0170477 (30:715)
8.: G. Dahlquist.
Error analysis for a class of methods for stiff non-linear initial value problems.
In Numerical analysis (Proc. 6th Biennial Dundee Conf., Univ. Dundee, Dundee, 1975), pages 60-72. Lecture Notes in Math., Vol. 506. Springer, Berlin, 1976. MR 0448898 (56:7203)
9.: Weinan E.
Analysis of the heterogeneous multiscale method for ordinary differential equations.
Commun. Math. Sci., 1(3):423-436, 2003. MR 2069938 (2005f:65082)
10.: Weinan E and B. Engquist.
The heterogeneous multiscale methods.
Commun. Math. Sci., 1(1):87-132, 2003. MR 1979846 (2004b:35019)
11.: Weinan E, D. Liu, and E. Vanden-Eijnden.
Analysis of multiscale methods for stochastic differential equations.
Commun. on Pure and Applied Math., 58:1544-1585, 2005. MR 2165382 (2006g:60103)
12.: B. Engquist and Y.-H. Tsai.
Heterogeneous multiscale methods for stiff ordinary differential equations.
Math. Comp., 74(252):1707-1742, 2005. MR 2164093 (2006e:65111)
13.: I. Fatkullin and E. Vanden-Eijnden.
A computational strategy for multiscale chaotic systems with applications to Lorenz 96 model.
J. Comp. Phys., 200:605-638, 2004. MR 2095278 (2005e:65209)
14.: E. Fermi, J. Pasta, and S. Ulam.
Studies of the nonlinear problems, I.
Los Alamos Report LA-1940, 1955.
Later published in Collected Papers of Enrico Fermi, ed. E. Segre, Vol. II (University of Chicago Press, 1965) p. 978.
15.: B. Garcıa-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, 1999. MR 1648882 (99g:65087)
16.: W. Gautschi.
Numerical integration of ordinary differential equations based on trigonometric polynomials.
Numerische Mathematik, 3:381-397, 1961. MR 0138200 (25:1647)
17.: C. W. Gear and I. G. Kevrekidis.
Projective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum.
SIAM J. Sci. Comput., 24(4):1091-1106 (electronic), 2003. MR 1976207 (2004c:65065)
18.: C. W. Gear and K. A. Gallivan.
Automatic methods for highly oscillatory ordinary differential equations.
In Numerical analysis (Dundee, 1981), volume 912 of Lecture Notes in Math., pages 115-124. Springer, 1982. MR 654346 (83f:65108)
19.: E. Hairer, C. Lubich, and G. 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)
20.: E. Hairer and G. Wanner.
Solving ordinary differential equations. II, volume 14 of Springer Series in Computational Mathematics.
Springer-Verlag, 1996. MR 1439506 (97m:65007)
21.: M. Hochbruck, C. Lubich, and H. Selhofer.
Exponential integrators for large systems of differential equations.
SIAM J. Sci. Comp., 19:1552-1574, 1998. MR 1618808 (99f:65101)
22.: A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett, and A. Zanna.
Lie-group methods.
In Acta numerica, 2000, volume 9, pages 215-365. Cambridge Univ. Press, 2000. MR 1883629 (2003a:37123)
23.: J. Kevorkian and J. D. Cole.
Perturbation Methods in Applied Mathematics, volume 34 of Applied Mathematical Sciences.
Springer-Verlag, New York, Heidelberg, Berlin, 1980. MR 608029 (82g:34082)
24.: J. Kevorkian and J. D. Cole.
Multiple Scale and Singular Perturbation Methods, volume 114 of Applied Mathematical Sciences.
Springer-Verlag, New York, Berlin, Heidelberg, 1996. MR 1392475 (97k:34001)
25.: H.-O. Kreiss.
Difference methods for stiff ordinary differential equations.
SIAM J. Numer. Anal., 15(1):21-58, 1978. MR 486570 (80a:65149)
26.: H.-O. Kreiss.
Problems with different time scales.
In Acta numerica, 1992, pages 101-139. Cambridge Univ. Press, 1992. MR 1165724 (93e:65111)
27.: H.-O. Kreiss and J. Lorenz.
Manifolds of slow solutions for highly oscillatory problems.
Indiana Univ. Math. J., 42(4):1169-1191, 1993. MR 1266089 (95f:34077)
28.: J. Laskar.
Large scale chaos in the solar system.
Astron. Astrophys, 287, 1994.
29.: V. I. Lebedev and S. A. Finogenov.
The use of ordered Čebyšhev parameters in iteration methods.
Ž. Vyčisl. Mat. i Mat. Fiz., 16(4):895-907, 1084, 1976. MR 0443314 (56:1684)
30.: B. Leimkuhler and S. Reich.
Simulating Hamiltonian dynamics, volume 14 of Cambridge Monographs on Applied and Computational Mathematics.
Cambridge University Press, 2004. MR 2132573 (2006a:37078)
31.: M. Levi.
Geometry and physics of averaging with applications.
Physica D, 132:150-164, 1999. MR 1705702 (2000g:34072)
32.: R.E. O'Malley.
On nonlinear singularly perturbed initial value problems.
SIAM Review, 30(2):193-212, 1988. MR 941110 (89g:65089)
33.: R. L. Petzold, O. J. Laurent, and Y. Jeng.
Numerical solution of highly oscillatory ordinary differential equations.
Acta Numerica, 6:437-483, 1997. MR 1489260 (98k:65040)
34.: J. A. Sanders and F. Verhulst.
Averaging Methods in Nonlinear Dynamical Systems, volume 59 of Applied Mathematical Sciences.
Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985. MR 810620 (87d:34065)
35.: R. E. Scheid.
The accurate numerical solution of highly oscillatory ordinary differential equations.
Mathematics of Computation, 41(164):487-509, 1983. MR 717698 (85i:65091)
36.: R. Sharp, Y.-H. Tsai, and B. Engquist.
Multiple time scale numerical methods for the inverted pendulum problem.
In Multiscale methods in science and engineering, volume 44 of Lect. Notes Comput. Sci. Eng., pages 241-261. Springer, Berlin, 2005. MR 2161717 (2006c:70002)
37.: D. Ter Haar, editor.
Collected Papers of P.L. Kapitza, volume II.
Pergamon Press, 1965.
38.: E. Vanden-Eijnden.
Numerical techniques for multi-scale dynamical systems with stochastic effects.
Comm. Math. Sci., 1:385-391, 2003. MR 1980483

Retrieve articles in Mathematics of Computation with MSC (2000): 65L05, 34E13, 34E20

Gil Ariel
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
Email: ariel@math.utexas.edu

Bjorn Engquist
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
Email: engquist@math.utexas.edu

Richard Tsai
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
Email: ytsai@math.utexas.edu

DOI: 10.1090/S0025-5718-08-02139-X
PII: S 0025-5718(08)02139-X
Received by editor(s): June 19, 2007
Received by editor(s) in revised form: January 20, 2008
Posted: October 3, 2008
Dedicated: In Memory of Germund Dahlquist
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS