Skip to main content
SpringerLink
Log in

A Multiscale Method for Highly Oscillatory Dynamical Systems Using a Poincaré Map Type Technique

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We propose a new heterogeneous multiscale method (HMM) for computing the effective behavior of a class of highly oscillatory ordinary differential equations (ODEs). Without the need for identifying hidden slow variables, the proposed method is constructed based on the following ideas: a nonstandard splitting of the vector field (the right hand side of the ODEs); comparison of the solutions of the split equations; construction of effective paths in the state space whose projection onto the slow subspace has the correct dynamics; and a novel on-the-fly filtering technique for achieving a high order accuracy. Numerical examples are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Ariel, G., Vanden-Eijnden, E.: Accelerated simulation of a heavy particle in a gas of elastic spheres. Multiscale Model. Simul. 7(1), 349–361 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ariel, G., Engquist, B., Tsai, R.: Numerical multiscale methods for coupled oscillators. Multiscale Model. Simul. 7(3), 1387–1404 (2008)

    Article  MathSciNet  Google Scholar 

  3. Ariel, G., Engquist, B., Kreiss, H.-O., Tsai, R.: Multiscale computations for highly oscillatory problems. In: Multiscale Modeling and Simulation in Science. Lect. Notes Comput. Sci. Eng., vol. 66, pp. 237–287. Springer, Berlin (2009)

    Chapter  Google Scholar 

  4. Ariel, G., Engquist, B., Tsai, R.: A multiscale method for highly oscillatory ordinary differential equations with resonance. Math. Comput. 78, 929–956 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Ariel, G., Engquist, B., Tsai, R.: A reversible multiscale integration method. Commun. Math. Sci. 7(3), 595–610 (2009)

    MathSciNet  MATH  Google Scholar 

  6. Ariel, G., Engquist, B., Tsai, R.: Oscillatory systems with three separated time scales: analysis and computation. In: Numerical Analysis of Multiscale Computations. Lecture Notes in Computational Science and Engineering, vol. 82, pp. 23–45. Springer, Berlin (2012)

    Chapter  Google Scholar 

  7. Ariel, G., Sanz-Serna, J.M., Tsai, R.: A multiscale technique for finding slow manifolds of stiff mechanical systems. Multiscale Model. Simul. 10(4), 1180–1203 (2012)

    Article  Google Scholar 

  8. Artstein, Z., Kevrekidis, I.G., Slemrod, M., Titi, E.S.: Slow observables of singularly perturbed differential equations. Nonlinearity 20(11), 2463–2481 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Artstein, Z., Linshiz, J., Titi, E.S.: Young measure approach to computing slowly advancing fast oscillations. Multiscale Model. Simul. 6(4), 1085–1097 (2007)

    Article  MathSciNet  Google Scholar 

  10. Bambusi, D., Ponno, A.: On metastability in FPU. Commun. Math. Phys. 264(2), 539–561 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bambusi, D., Ponno, A.: Resonance, metastability and blow up in FPU. In: The Fermi-Pasta-Ulam Problem. Lecture Notes in Phys., vol. 728, pp. 191–205. Springer, Berlin (2008)

    Chapter  Google Scholar 

  12. Bond, S.D., Leimkuhler, B.J.: Molecular dynamics and the accuracy of numerically computed averages. Acta Numer. 16, 1–65 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chu, J., Engquist, B., Prodanovic, M., Tsai, R.: A multiscale method coupling network and continuum models in porous media I—Single phase flow. Multiscale Model Simul. 10(2), 515–549 (2012)

    Article  MATH  Google Scholar 

  14. Chu, J., Engquist, B., Prodanovic, M., Tsai, R.: A multiscale method coupling network and continuum models in porous media II—Single and two phase flow. In: Advances in Applied Mathematics, Modeling, and Computational Science. Fields Institute Communications, vol. 66, pp. 161–185 (2013)

    Chapter  Google Scholar 

  15. Cohen, D., Jahnke, T., Lorenz, K., Lubich, C.: Numerical integrators for highly oscillatory Hamiltonian systems: a review. In: Analysis, Modeling and Simulation of Multiscale Problems, pp. 553–576. Springer, Berlin (2006)

    Chapter  Google Scholar 

  16. Condon, M., Deaño, A., Iserles, A.: On second-order differential equations with highly oscillatory forcing terms. Proc. R. Soc. Lond. Ser. A. Math. Phys. Eng. Sci. 466(2118), 1809–1828 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Deuflhard, P.: A study of extrapolation methods based on multistep schemes without parasitic solutions. Z. Angew. Math. Phys. 30(2), 177–189 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  18. E, W.: Analysis of the heterogeneous multiscale method for ordinary differential equations. Commun. Math. Sci. 1(3), 423–436 (2003)

    MathSciNet  MATH  Google Scholar 

  19. E, W., Engquist, B.: The heterogeneous multiscale methods. Commun. Math. Sci. 1(1), 87–132 (2003)

    MathSciNet  MATH  Google Scholar 

  20. E, W., Vanden-Eijnden, E.: Numerical techniques for multiscale dynamical systems with stochastic effects. Commun. Math. Sci. 1(2), 385–391 (2003)

    MathSciNet  Google Scholar 

  21. E, W., Engquist, B., Li, X., Ren, W., Vanden-Eijnden, E.: Heterogeneous multiscale methods: a review. Commun. Comput. Phys. 2(3), 367–450 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Engquist, B., Tsai, Y.-H.: Heterogeneous multiscale methods for stiff ordinary differential equations. Math. Comput. 74(252), 1707–1742 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Engquist, B., Holst, H., Runborg, O.: Multi-scale methods for wave propagation in heterogeneous media over long time. In: Engquist, B., Runborg, O., Tsai, R. (eds.) Numerical Analysis of Multiscale Computations. Lect. Notes Comput. Sci. Eng., vol. 82. Springer, Berlin (2011)

    Google Scholar 

  24. Fatkullin, I., Vanden-Eijnden, E.: A computational strategy for multiscale chaotic systems with applications to Lorenz 96 model. J. Comput. Phys. 200, 605–638 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. García-Archilla, B., Sanz-Serna, J.M., Skeel, R.D.: Long-time-step methods for oscillatory differential equations. SIAM J. Sci. Comput. 20(3), 930–963 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gear, C.W., Kevrekidis, I.G.: Projective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum. SIAM J. Sci. Comput. 24(4), 1091–1106 (2003). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gear, C.W., Kevrekidis, I.G.: Constraint-defined manifolds: a legacy code approach to low-dimensional computation. J. Sci. Comput. 25(1–2), 17–28 (2005)

    MathSciNet  MATH  Google Scholar 

  28. Hairer, E., Lubich, C.: On the energy distribution in Fermi-Pasta-Ulam lattices. Arch. Ration. Mech. Anal. 205(3), 993–1029 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer Series in Computational Mathematics, vol. 31. Springer, Berlin (2002)

    MATH  Google Scholar 

  30. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Iserles, A., Munthe-Kaas, H.Z., Nørsett, S.P., Zanna, A.: Lie-group methods. In: Acta Numerica, 2000. Acta Numer., vol. 9, pp. 215–365. Cambridge Univ. Press, Cambridge (2000)

    Google Scholar 

  32. Kevorkian, J., Cole, J.D.: Perturbation Methods in Applied Mathematics. Applied Mathematical Sciences, vol. 34. Springer, New York (1981)

    Book  MATH  Google Scholar 

  33. Kevorkian, J., Cole, J.D.: Multiple Scale and Singular Perturbation Methods. Applied Mathematical Sciences, vol. 114. Springer, New York (1996)

    Book  MATH  Google Scholar 

  34. Kreiss, H.-O.: Problems with different time scales for ordinary differential equations. SIAM J. Numer. Anal. 16(6), 980–998 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kreiss, H.-O.: Problems with different time scales. In: Acta Numerica, 1992, pp. 101–139. Cambridge Univ. Press, Cambridge (1992)

    Google Scholar 

  36. Kreiss, H.-O., Lorenz, J.: Manifolds of slow solutions for highly oscillatory problems. Indiana Univ. Math. J. 42(4), 1169–1191 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  37. Leimkuhler, B., Reich, S.: Simulating Hamiltonian Dynamics. Cambridge Monographs on Applied and Computational Mathematics, vol. 14. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  38. Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer., 357–514 (2001)

  39. Petzold, L.R., Jay, L.O., Yen, J.: Numerical solution of highly oscillatory ordinary differential equations. In: Acta Numerica, 1997. Acta Numer., vol. 6, pp. 437–483. Cambridge Univ. Press, Cambridge (1997)

    Google Scholar 

  40. Sanders, J.A., Verhulst, F.: Averaging Methods in Nonlinear Dynamical Systems. Applied Mathematical Sciences, vol. 59. Springer, New York (1985)

    MATH  Google Scholar 

  41. Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems. Applied Mathematics and Mathematical Computation, vol. 7. Chapman & Hall, London (1994)

    MATH  Google Scholar 

  42. Tao, M., Owhadi, H., Marsden, J.E.: Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging. Multiscale Model. Simul. 8(4), 1269–1324 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  43. Vanden-Eijnden, E.: Numerical techniques for multi-scale dynamical systems with stochastic effects. Commun. Math. Sci. 1, 385–391 (2003)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Kim and Tsai are partially supported by NSF grants DMS-0914840 and DMS-0914465. Engquist was partially supported by NSF grant DMS-0714612.

Author information

Authors and Affiliations

  1. Bar-Ilan University, Ramat Gan, 52900, Israel

    G. Ariel

  2. Department of Mathematics and Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin, TX, 78712, USA

    B. Engquist, S. Kim, Y. Lee & R. Tsai

Authors
  1. G. Ariel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Engquist
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Y. Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. R. Tsai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Tsai.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ariel, G., Engquist, B., Kim, S. et al. A Multiscale Method for Highly Oscillatory Dynamical Systems Using a Poincaré Map Type Technique. J Sci Comput 54, 247–268 (2013). https://doi.org/10.1007/s10915-012-9656-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-012-9656-x

Keywords

Search

Navigation