Hopf bifurcation of reaction-diffusion and Navier-Stokes equations under discretization |
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Authors: | Christian Lubich Alexander Ostermann |
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Institution: | Mathematisches Institut, Universit?t Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany; e-mail: lubich@na.uni-tuebingen.de, DE Institut für Mathematik und Geometrie, Universit?t Innsbruck, Technikerstra?e 13, A-6020 Innsbruck, Austria; e-mail: alex@mat1.uibk.ac.at, AT
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Abstract: | Summary. The long-time behaviour of numerical approximations to the solutions of a semilinear parabolic equation undergoing a Hopf
bifurcation is studied in this paper. The framework includes reaction-diffusion and incompressible Navier-Stokes equations.
It is shown that the phase portrait of a supercritical Hopf bifurcation is correctly represented by Runge-Kutta time discretization.
In particular, the bifurcation point and the Hopf orbits are approximated with higher order. A basic tool in the analysis
is the reduction of the dynamics to a two-dimensional center manifold. A large portion of the paper is therefore concerned
with studying center manifolds of the discretization.
Received March 18, 1997 / Revised version received February 19, 1998 |
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Keywords: | Mathematics Subject Classification (1991):65M12 65M15 35B32 58F14 |
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