Hopf bifurcation of reaction-diffusion and Navier-Stokes equations under discretization (original) (raw)
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.
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Authors and Affiliations
- Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany; e-mail: lubich@na.uni-tuebingen.de , , , , , , DE
Christian Lubich - Institut für Mathematik und Geometrie, Universität Innsbruck, Technikerstraße 13, A-6020 Innsbruck, Austria; e-mail: alex@mat1.uibk.ac.at , , , , , , AT
Alexander Ostermann
Authors
- Christian Lubich
- Alexander Ostermann
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Received March 18, 1997 / Revised version received February 19, 1998
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Lubich, C., Ostermann, A. Hopf bifurcation of reaction-diffusion and Navier-Stokes equations under discretization.Numer. Math. 81, 53–84 (1998). https://doi.org/10.1007/s002110050384
- Issue date: November 1998
- DOI: https://doi.org/10.1007/s002110050384