The Global Attractor Under Discretisation

  • Andrew Stuart
Part of the NATO ASI Series book series (ASIC, volume 313)

Abstract

The effect of temporal discretisation on dissipative differential equations is analysed. We discuss the effect of discretisation on the global attractor and survey some recent results in the area. The advantage of concentrating on ω and α limit sets (which are contained in the global attractor) is described. An analysis of spurious bifurcations in the ω and α limit sets is presented for linear multistep methods, using the time-step Δtas the bifurcation parameter. The results arising from application of local bifurcation theory are shown to hold globally and a necessary and sufficient condition is derived for the non-existence of a particular class of spurious solutions, for allΔt> 0. The class of linear multistep methods satisfying this condition is fairly restricted since the underlying theory is very general and takes no account of any inherent structure in the underlying differential equations. Hence a method complementary to the bifurcation analysis is described, the aim being to construct methods for which spurious solutions do not exist forΔt sufficiently small; for infinite dimensional dynamical systems the method relies on examining steady boundary value problems (which govern the existence of spurious solutions) in the singular limit correpsonding to Δt→ 0+. The analysis we describe is helpful in the design of schemes for long-time simulations.

Keywords

Unstable Manifold Global Attractor Steady Solution Euler Scheme Linear Multistep Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    F.Brezzi, S.Ushiki, H.Fujii Real and ghost bifurcation dynamics in difference schemes for ordinary differential equations. Appears in Numerical Methods for Bifurcation Problems, Eds: T.Kupper, H.D.Mittleman, H.Weber. Birkhauser-Verlag, Boston, 1984.Google Scholar
  2. [2]
    K.Burrage, J.C.Butcher Stability criteria for implicit Runge-Kutta methods. SIAM J. Num. Anal. 16 (1979), 46–57.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    N.Chafee, E.Infante A bifurcation problem for a nonlinear parabolic equation. J. Applic. Anal., 4 (1974), 17–37.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    S.N.Chow, J.K.Hale Methods of Bifurcation Theory. Springer Verlag, New York, 1982.MATHGoogle Scholar
  5. [5]
    C.M.ElliottIn preparationGoogle Scholar
  6. [6]
    E.Hairer, A.Iserles, J.M.Sanz-Serna Equilibria of Runge-Kutta methods. University of Cambridge, DAMTP Report 1989/NA4.Google Scholar
  7. [7]
    J.K.Hale, L.T.Magalhaes, W.M.Oliva An Introduction to Infinite Dimensional Dynamical Sytems - Geometric Theory. Springer Lecture Notes in Applied Mathematics, 47. 1984.Google Scholar
  8. [8]
    J.K.Hale, X.-B.Lin, G.Raugel Upper semicontinuity of attractors for approximations of semigroups and partial differential equations. Math. Comp. 181 (1988), 89–123.MathSciNetCrossRefGoogle Scholar
  9. [9]
    J.K.Hale Asymptotic Behaviour of Dissipative Dynamical Systems. AMS Mathematical Surveys and Monographs # 25, 1988.Google Scholar
  10. [10]
    A.Iserles Stability and dynamics of numerical methods for nonlinear ordinary differential equations. IMA J. Num. Anal. 10 (1990), 1–30.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    A.Iserles, A.T.Peplow, A.M.Stuart A unified approach to spurious solutions introduced by time discretisation. In preparation.Google Scholar
  12. [12]
    P.E.Kloeden, J.Lorenz Stable attracting sets in dynamical systems and in their one-step discretisationsSIAM J. Num. Anal. 23 (1986), 986–995.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    P.E.Kloeden, J.Lorenz A note on multistep methods and attracting sets of dynamical systems. To appear in Numer. Math.Google Scholar
  14. [14]
    A.M.Stuart Nonlinear instability in dissipative finite difference schemes. SIAM Review 31 (1989), 191–220.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    A.M.Stuart Linear instability implies spurious periodic solutions. IMA J. Num. Anal. 9 (1989), 465–486.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    A.M.Stuart, A.T.Peplow The dynamics of the theta method. Submitted to SIAM J. Sci. Stat. Comp.Google Scholar
  17. [17]
    R.Temam Infinite Dimnsional Dynamical Systems in Mathematics and Physics. Applied Mathematical Science. # 68, Springer-Verlag, 1988.Google Scholar

Copyright information

© Kluwer Academic Publishers 1990

Authors and Affiliations

  • Andrew Stuart
    • 1
  1. 1.School of Mathematical SciencesUniversity of BathBathUK

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