Journal ArticleDOI
Numerical Approximation of Relative Equilibria for Equivariant PDEs
TLDR
Convergence results for the numerical approximation of relative equilibria of parabolic systems in one space dimension are proved.Abstract:
We prove convergence results for the numerical approximation of relative equilibria of parabolic systems in one space dimension. These systems are special examples of equivariant evolution equations. We use finite differences on a large interval with appropriately chosen boundary conditions. Moreover, we consider the approximation of isolated eigenvalues of finite multiplicity of the linear operator which arises from linearization at the equilibrium as well as the approximation of the corresponding invariant subspace. The results in this paper which are a generalization of the results in [V. Thummler, Numerical Analysis of the Method of Freezing Traveling Waves, Ph.D. thesis, Bielefeld University, 2005] are illustrated by numerical computations for the cubic quintic Ginzburg-Landau equation.read more
Citations
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Reduction and reconstruction for self-similar dynamical systems
TL;DR: In this paper, the authors present a general method for analysing and numerically solving partial differential equations with self-similar solutions, which employs ideas from symmetry reduction in geometric mechanics and involves separating the dynamics on the shape space from those on the group space.
Book ChapterDOI
Phase Conditions, Symmetries and PDE Continuation
Wolf-Jürgen Beyn,Vera Thümmler +1 more
TL;DR: In this article, the authors discuss the usefulness of phase conditions for the numerical analysis of finite and infinite-dimensional dynamical systems that have continuous symmetries and present an abstract framework for evolution equations that are equivariant with respect to the action of a (not necessarily compact) Lie group.
Journal ArticleDOI
The Large Core Limit of Spiral Waves in Excitable Media: A Numerical Approach
TL;DR: In this article, the authors modify the freezing method introduced by Beyn and Thummler in 2004 for analyzing rigidly rotating spiral waves in excitable media, and propose boundary conditions based on geometric approximations of spiral wave solutions by Archimedean spirals and by involutes of circles.
Book ChapterDOI
Stability and Computation of Dynamic Patterns in PDEs
TL;DR: In this article, the authors give an overview of problems related to the theoretical and numerical analysis of nonlinear wave patterns and relate it to linearized stability determined by the spectral behavior of linearized operators.
Journal ArticleDOI
The Effect of Freezing and Discretization to the Asymptotic Stability of Relative Equilibria
TL;DR: In this paper, nonlinear stability results for the numerical approximation of relative equilibria of equivariant parabolic partial differential equations in one space dimension were proved. But the main result is that the stability properties are inherited by numerical approximation with finite differences on a finite equidistant grid with appropriate boundary conditions.
References
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Book
Perturbation theory for linear operators
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
Book
Applications of Lie Groups to Differential Equations
TL;DR: In this paper, the Cauchy-Kovalevskaya Theorem has been used to define a set of invariant solutions for differential functions in a Lie Group.
Book
Geometric Theory of Semilinear Parabolic Equations
TL;DR: The neighborhood of an invariant manifold near an equilibrium point is a neighborhood of nonlinear parabolic equations in physical, biological and engineering problems as mentioned in this paper, where the neighborhood of a periodic solution is defined by the invariance of the manifold.
BookDOI
Introduction to mechanics and symmetry
TL;DR: A basic exposition of classical mechanical systems; 2nd edition Reference CAG-BOOK-2008-008 Record created on 2008-11-21, modified on 2017-09-27 as mentioned in this paper.