Efficient numerical algorithms for the continuation of periodic orbits of high-dimensional dissipative dynamical systems, and for analyzing their stability are presented. They are based on shooting, Newton–Krylov and Arnoldi methods. A thermal convection fluid dynamics problem, which has a rich bifurcation diagram due to symmetries, has been used as test. After a pseudo-spectral discretization of the equations a system of dimension O(104) has been obtained. The efficiency of the algorithms, which allows the unfolding of a complex diagram of periodic orbits, makes the methods suitable for the study of large nonlinear dissipative partial differential equations.
Description
Newton–Krylov continuation of periodic orbits for Navier–Stokes flows - ScienceDirect
%0 Journal Article
%1 snchez2004newtonkrylov
%A Sánchez, J.
%A Net, M.
%A Garcı́a-Archilla, B.
%A Simó, C.
%D 2004
%J Journal of Computational Physics
%K 37l15-stability-problems-for-infinite-dimensional-dissipative-systems 37n10-dynamical-systems-in-fluid-mechanics-oceanography-fluid-mechanics 65p30-numerical-problems-in-dynamical-systems-bifurcation 76d05-incompressible-navier-stokes-equations
%N 1
%P 13 - 33
%R 10.1016/j.jcp.2004.04.018
%T Newton–Krylov continuation of periodic orbits for Navier–Stokes flows
%U http://www.sciencedirect.com/science/article/pii/S0021999104001895
%V 201
%X Efficient numerical algorithms for the continuation of periodic orbits of high-dimensional dissipative dynamical systems, and for analyzing their stability are presented. They are based on shooting, Newton–Krylov and Arnoldi methods. A thermal convection fluid dynamics problem, which has a rich bifurcation diagram due to symmetries, has been used as test. After a pseudo-spectral discretization of the equations a system of dimension O(104) has been obtained. The efficiency of the algorithms, which allows the unfolding of a complex diagram of periodic orbits, makes the methods suitable for the study of large nonlinear dissipative partial differential equations.
@article{snchez2004newtonkrylov,
abstract = {Efficient numerical algorithms for the continuation of periodic orbits of high-dimensional dissipative dynamical systems, and for analyzing their stability are presented. They are based on shooting, Newton–Krylov and Arnoldi methods. A thermal convection fluid dynamics problem, which has a rich bifurcation diagram due to symmetries, has been used as test. After a pseudo-spectral discretization of the equations a system of dimension O(104) has been obtained. The efficiency of the algorithms, which allows the unfolding of a complex diagram of periodic orbits, makes the methods suitable for the study of large nonlinear dissipative partial differential equations.},
added-at = {2021-01-28T01:23:00.000+0100},
author = {Sánchez, J. and Net, M. and Garcı́a-Archilla, B. and Simó, C.},
biburl = {https://www.bibsonomy.org/bibtex/2f9840d8f2e8ebf06bbc7f23f09eb0c98/gdmcbain},
description = {Newton–Krylov continuation of periodic orbits for Navier–Stokes flows - ScienceDirect},
doi = {10.1016/j.jcp.2004.04.018},
interhash = {f739b99f3b6d1a00db0b621413c9361f},
intrahash = {f9840d8f2e8ebf06bbc7f23f09eb0c98},
issn = {0021-9991},
journal = {Journal of Computational Physics},
keywords = {37l15-stability-problems-for-infinite-dimensional-dissipative-systems 37n10-dynamical-systems-in-fluid-mechanics-oceanography-fluid-mechanics 65p30-numerical-problems-in-dynamical-systems-bifurcation 76d05-incompressible-navier-stokes-equations},
number = 1,
pages = {13 - 33},
timestamp = {2021-01-28T01:24:28.000+0100},
title = {Newton–Krylov continuation of periodic orbits for Navier–Stokes flows},
url = {http://www.sciencedirect.com/science/article/pii/S0021999104001895},
volume = 201,
year = 2004
}