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Nonlinear Galerkin Methods For The 3D Magnetohydrodynamic Equations (1997)
| Venue: | Int. J. Bifurcation and Chaos |
| Citations: | 3 - 2 self |
BibTeX
@ARTICLE{Schmidtmann97nonlineargalerkin,
author = {Olaf Schmidtmann and Fred Feudel and Norbert Seehafer},
title = {Nonlinear Galerkin Methods For The 3D Magnetohydrodynamic Equations},
journal = {Int. J. Bifurcation and Chaos},
year = {1997},
pages = {7--1497}
}
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Abstract
The usage of nonlinear Galerkin methods for the numerical solution of partial differential equations is demonstrated by treating an example. We desribe the implementation of a nonlinear Galerkin method based on an approximate inertial manifold for the 3D magnetohydrodynamic equations and compare its efficiency with the linear Galerkin approximation. Special bifurcation points, time-averaged values of energy and enstrophy as well as Kaplan-- Yorke dimensions are calculated for both schemes in order to estimate the number of modes necessary to correctly describe the behavior of the exact solutions. 1 Introduction Nonlinear Galerkin methods are new numerical methods which have been introduced recently by Temam and Marion in connection with the theory of inertial manifolds (IMs) and approximate inertial manifolds (AIMs) to study the long-term behavior of dissipative partial differential equations (PDEs) (see e.g., Marion & Temam [1989]). The essential aim when using these methods is to ch...
Keyphrases
nonlinear galerkin method magnetohydrodynamic equation approximate inertial manifold introduction nonlinear galerkin method numerical solution essential aim linear galerkin approximation partial differential equation exact solution dissipative partial differential equation long-term behavior kaplan yorke dimension time-averaged value marion temam new numerical method special bifurcation point inertial manifold
