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Preserving dissipation in approximate inertial forms for the Kuramoto- Sivashinsky equation. (English) Zbl 0738.35024

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 pints and inaccurate solution branches. These shortcomings were attributed to a lack of dissipation in the forms used. The authors show analytically and verity numerically that with an appropriate adjustment they can eliminate these numerical artifacts. The motivation for this adjustment is to enforce dissipation, while maintaining the same order of approximation. It is demonstrated 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, it is shown that certain implicit forms are dissipative without any adjustment. As an illustrative example the Kuramoto-Sivashinsky equation is used.

MSC:

35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B32 Bifurcations in context of PDEs
58D25 Equations in function spaces; evolution equations
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