Journal of Dynamics and Differential Equations

, Volume 3, Issue 2, pp 179–197

Preserving dissipation in approximate inertial forms for the Kuramoto-Sivashinsky equation


  • M. S. Jolly
    • Program in Applied and Computational MathematicsPrinceton University
    • Department of MathematicsIndiana University
  • I. G. Kevrekidis
    • Program in Applied and Computational MathematicsPrinceton University
    • Department of Chemical EngineeringPrinceton University
  • E. S. Titi
    • Mathematical Sciences InstituteCornell University
    • Department of MathematicsUniversity of California

DOI: 10.1007/BF01047708

Cite this article as:
Jolly, M.S., Kevrekidis, I.G. & Titi, E.S. J Dyn Diff Equat (1991) 3: 179. doi:10.1007/BF01047708


It has been observed, in earlier computations of bifurcation diagrams for dissipative partial differential equations, that the use of certain explicit approximate inertial forms can give rise to numerical artifacts such as spurious turning points and inaccurate solution branches. These shortcomings were attributed to a lack of dissipation in the forms used. We show analytically and verify numerically that with an appropriate adjustment we can eliminate these numerical artifacts. The motivation for this adjustment is to enforce dissipation, while maintaining the same order of approximation. We demonstrate with computations that the most natural remedy, namely, preparation of the equation, can be highly sensitive to assumptions on the size of the absorbing ball. In addition, we show that certain implicit forms are dissipative without any adjustment. As an illustrative example we use here the Kuramoto-Sivashinsky equation.

Key words

Inertial manifolddissipationKuramoto-Sivashinskynon-linear Galerkin method
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Copyright information

© Plenum Publishing Corporation 1991