A multiscale method for highly oscillatory ordinary differential equations with resonance

Ariel, Gil; Engquist, Bjorn; Tsai, Richard

doi:10.1090/S0025-5718-08-02139-X

A multiscale method for highly oscillatory ordinary differential equations with resonance
HTML articles powered by AMS MathViewer

by Gil Ariel, Bjorn Engquist and Richard Tsai PDF

Math. Comp. 78 (2009), 929-956 Request permission

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.

References

Assyr Abdulle, Fourth order Chebyshev methods with recurrence relation, SIAM J. Sci. Comput. 23 (2002), no. 6, 2041–2054. MR 1923724, DOI 10.1137/S1064827500379549
G. Ariel, B. Engquist, and R. Tsai. A multiscale method for weakly coupled oscillators. Submitted to Multi. Model. Simul.
N. N. Bogoliubov and Yu. A. Mitropolski. Asymptotic Methods in the Theory of Nonlinear Oscillations. Gordon and Breach, New York, 1961.
G. L. Browning and H.-O. Kreiss, Multiscale bounded derivative initialization for an arbitrary domain, J. Atmospheric Sci. 59 (2002), no. 10, 1680–1696. MR 1905843, DOI 10.1175/1520-0469(2002)059<1680:MBDIFA>2.0.CO;2
Germund Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand. 4 (1956), 33–53. MR 80998, DOI 10.7146/math.scand.a-10454
Germund Dahlquist, Stability and error bounds in the numerical integration of ordinary differential equations, Kungl. Tekn. Högsk. Handl. Stockholm 130 (1959), 87. MR 102921
Germund G. Dahlquist, A special stability problem for linear multistep methods, Nordisk Tidskr. Informationsbehandling (BIT) 3 (1963), 27–43. MR 170477, DOI 10.1007/bf01963532
Germund Dahlquist, Error analysis for a class of methods for stiff non-linear initial value problems, Numerical analysis (Proc. 6th Biennial Dundee Conf., Univ. Dundee, Dundee, 1975) Lecture Notes in Math., Vol. 506, Springer, Berlin, 1976, pp. 60–72. MR 0448898
Weinan E, Analysis of the heterogeneous multiscale method for ordinary differential equations, Commun. Math. Sci. 1 (2003), no. 3, 423–436. MR 2069938
Weinan E and Bjorn Engquist, The heterogeneous multiscale methods, Commun. Math. Sci. 1 (2003), no. 1, 87–132. MR 1979846
Weinan E, Di Liu, and Eric Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Comm. Pure Appl. Math. 58 (2005), no. 11, 1544–1585. MR 2165382, DOI 10.1002/cpa.20088
Bjorn Engquist and Yen-Hsi Tsai, Heterogeneous multiscale methods for stiff ordinary differential equations, Math. Comp. 74 (2005), no. 252, 1707–1742. MR 2164093, DOI 10.1090/S0025-5718-05-01745-X
Ibrahim Fatkullin and Eric Vanden-Eijnden, A computational strategy for multiscale systems with applications to Lorenz 96 model, J. Comput. Phys. 200 (2004), no. 2, 605–638. MR 2095278, DOI 10.1016/j.jcp.2004.04.013
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.
B. García-Archilla, J. M. Sanz-Serna, and R. D. Skeel, Long-time-step methods for oscillatory differential equations, SIAM J. Sci. Comput. 20 (1999), no. 3, 930–963. MR 1648882, DOI 10.1137/S1064827596313851
Walter Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials, Numer. Math. 3 (1961), 381–397. MR 138200, DOI 10.1007/BF01386037
C. W. Gear and Ioannis G. Kevrekidis, Projective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum, SIAM J. Sci. Comput. 24 (2003), no. 4, 1091–1106. MR 1976207, DOI 10.1137/S1064827501388157
C. W. Gear and K. A. Gallivan, Automatic methods for highly oscillatory ordinary differential equations, Numerical analysis (Dundee, 1981) Lecture Notes in Math., vol. 912, Springer, Berlin-New York, 1982, pp. 115–124. MR 654346
Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric numerical integration, Springer Series in Computational Mathematics, vol. 31, Springer-Verlag, Berlin, 2002. Structure-preserving algorithms for ordinary differential equations. MR 1904823, DOI 10.1007/978-3-662-05018-7
E. Hairer and G. Wanner, Solving ordinary differential equations. II, 2nd ed., Springer Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin, 1996. Stiff and differential-algebraic problems. MR 1439506, DOI 10.1007/978-3-642-05221-7
Marlis Hochbruck, Christian Lubich, and Hubert Selhofer, Exponential integrators for large systems of differential equations, SIAM J. Sci. Comput. 19 (1998), no. 5, 1552–1574. MR 1618808, DOI 10.1137/S1064827595295337
Arieh Iserles, Hans Z. Munthe-Kaas, Syvert P. Nørsett, and Antonella Zanna, Lie-group methods, Acta numerica, 2000, Acta Numer., vol. 9, Cambridge Univ. Press, Cambridge, 2000, pp. 215–365. MR 1883629, DOI 10.1017/S0962492900002154
J. Kevorkian and Julian D. Cole, Perturbation methods in applied mathematics, Applied Mathematical Sciences, vol. 34, Springer-Verlag, New York-Berlin, 1981. MR 608029
J. Kevorkian and J. D. Cole, Multiple scale and singular perturbation methods, Applied Mathematical Sciences, vol. 114, Springer-Verlag, New York, 1996. MR 1392475, DOI 10.1007/978-1-4612-3968-0
Heinz-Otto Kreiss, Difference methods for stiff ordinary differential equations, SIAM J. Numer. Anal. 15 (1978), no. 1, 21–58. MR 486570, DOI 10.1137/0715003
Heinz-Otto Kreiss, Problems with different time scales, Acta numerica, 1992, Acta Numer., Cambridge Univ. Press, Cambridge, 1992, pp. 101–139. MR 1165724, DOI 10.1017/s0962492900002257
Heinz-Otto Kreiss and Jens Lorenz, Manifolds of slow solutions for highly oscillatory problems, Indiana Univ. Math. J. 42 (1993), no. 4, 1169–1191. MR 1266089, DOI 10.1512/iumj.1993.42.42054
J. Laskar. Large scale chaos in the solar system. Astron. Astrophys, 287, 1994.
V. I. Lebedev and S. A. Finogenov, The use of ordered Čebyšev parameters in iteration methods, Ž. Vyčisl. Mat i Mat. Fiz. 16 (1976), no. 4, 895–907, 1084 (Russian). MR 443314
Benedict Leimkuhler and Sebastian Reich, Simulating Hamiltonian dynamics, Cambridge Monographs on Applied and Computational Mathematics, vol. 14, Cambridge University Press, Cambridge, 2004. MR 2132573
Mark Levi, Geometry and physics of averaging with applications, Phys. D 132 (1999), no. 1-2, 150–164. MR 1705702, DOI 10.1016/S0167-2789(99)00022-6
R. E. O’Malley Jr., On nonlinear singularly perturbed initial value problems, SIAM Rev. 30 (1988), no. 2, 193–212. MR 941110, DOI 10.1137/1030044
Linda R. Petzold, Laurent O. Jay, and Jeng Yen, Numerical solution of highly oscillatory ordinary differential equations, Acta numerica, 1997, Acta Numer., vol. 6, Cambridge Univ. Press, Cambridge, 1997, pp. 437–483. MR 1489260, DOI 10.1017/S0962492900002750
J. A. Sanders and F. Verhulst, Averaging methods in nonlinear dynamical systems, Applied Mathematical Sciences, vol. 59, Springer-Verlag, New York, 1985. MR 810620, DOI 10.1007/978-1-4757-4575-7
Robert E. Scheid Jr., The accurate numerical solution of highly oscillatory ordinary differential equations, Math. Comp. 41 (1983), no. 164, 487–509. MR 717698, DOI 10.1090/S0025-5718-1983-0717698-9
Richard Sharp, Yen-Hsi Tsai, and Björn Engquist, Multiple time scale numerical methods for the inverted pendulum problem, Multiscale methods in science and engineering, Lect. Notes Comput. Sci. Eng., vol. 44, Springer, Berlin, 2005, pp. 241–261. MR 2161717, DOI 10.1007/3-540-26444-2_{1}3
D. Ter Haar, editor. Collected Papers of P.L. Kapitza, volume II. Pergamon Press, 1965.
Eric Vanden-Eijnden, Numerical techniques for multi-scale dynamical systems with stochastic effects, Commun. Math. Sci. 1 (2003), no. 2, 385–391. MR 1980483