We provide an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems. Many of these problems are characterized by high-dimensional dynamical systems which undergo transitions as parameters are changed. The computation of the critical conditions associated with these transitions, popularly referred to as `tipping points', is important for understanding the transition mechanisms. We describe the two basic classes of methods of numerical bifurcation analysis, which differ in the explicit or implicit use of the Jacobian matrix of the dynamical system. The numerical challenges involved in both methods are mentioned and possible solutions to current bottlenecks are given. To demonstrate that numerical bifurcation techniques are not restricted to relatively low-dimensional dynamical systems, we provide several examples of the application of the modern techniques to a diverse set of fluid mechanical problems.
%0 Journal Article
%1 dijkstra2014numerical
%A Dijkstra, Henk A.
%A Wubs, Fred W.
%A Cliffe, Andrew K.
%A Doedel, Eusebius
%A Dragomirescu, Ioana F.
%A Eckhardt, Bruno
%A Yu, Alexander
%A Hazel, Andrew L.
%A Lucarini, Valerio
%A Salinger, Andy G.
%A Phipps, Erik T.
%A Sanchez-Umbria, Juan
%A Schuttelaars, Henk
%A Tuckerman, Laurette S.
%A Thiele, Uwe
%D 2014
%J Communications in Computational Physics
%K 35b32-pdes-bifurcation 35b60-pdes-continuation-and-prolongation-of-solutions 35q35-pdes-in-connection-with-fluid-mechanics 37h20-random-dynamical-systems-bifurcation-theory 37m05-simulation-of-dynamical-systems 65h20-global-methods-including-homotopy-approaches 65p30-numerical-problems-in-dynamical-systems-bifurcation 76-02-fluid-mechanics-research-exposition 76e06-hydrodynamic-stability-convection 76e30-hydrodynamic-stability-nonlinear-effects
%N 1
%P 1 - 45
%R 10.4208/cicp.240912.180613a
%T Numerical Bifurcation Methods and their Application to Fluid Dynamics: Analysis beyond Simulation
%U https://www.cambridge.org/core/journals/communications-in-computational-physics/article/abs/numerical-bifurcation-methods-and-their-application-to-fluid-dynamics-analysis-beyond-simulation/50CAA5AB617E7A23A44C6E90965F70A1
%V 15
%X We provide an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems. Many of these problems are characterized by high-dimensional dynamical systems which undergo transitions as parameters are changed. The computation of the critical conditions associated with these transitions, popularly referred to as `tipping points', is important for understanding the transition mechanisms. We describe the two basic classes of methods of numerical bifurcation analysis, which differ in the explicit or implicit use of the Jacobian matrix of the dynamical system. The numerical challenges involved in both methods are mentioned and possible solutions to current bottlenecks are given. To demonstrate that numerical bifurcation techniques are not restricted to relatively low-dimensional dynamical systems, we provide several examples of the application of the modern techniques to a diverse set of fluid mechanical problems.
@article{dijkstra2014numerical,
abstract = {We provide an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems. Many of these problems are characterized by high-dimensional dynamical systems which undergo transitions as parameters are changed. The computation of the critical conditions associated with these transitions, popularly referred to as `tipping points', is important for understanding the transition mechanisms. We describe the two basic classes of methods of numerical bifurcation analysis, which differ in the explicit or implicit use of the Jacobian matrix of the dynamical system. The numerical challenges involved in both methods are mentioned and possible solutions to current bottlenecks are given. To demonstrate that numerical bifurcation techniques are not restricted to relatively low-dimensional dynamical systems, we provide several examples of the application of the modern techniques to a diverse set of fluid mechanical problems.},
added-at = {2017-06-29T07:13:07.000+0200},
author = {Dijkstra, Henk A. and Wubs, Fred W. and Cliffe, Andrew K. and Doedel, Eusebius and Dragomirescu, Ioana F. and Eckhardt, Bruno and Yu, Alexander and Hazel, Andrew L. and Lucarini, Valerio and Salinger, Andy G. and Phipps, Erik T. and Sanchez-Umbria, Juan and Schuttelaars, Henk and Tuckerman, Laurette S. and Thiele, Uwe},
biburl = {https://www.bibsonomy.org/bibtex/202b55afd092d4f47493e60f4ffea7379/gdmcbain},
citeulike-article-id = {13836030},
citeulike-attachment-1 = {dijkstra_13_numerical.pdf; /pdf/user/gdmcbain/article/13836030/1042607/dijkstra_13_numerical.pdf; 6f06d893a2f244b0a3c50ba99a2f4114ffc5d8be},
citeulike-linkout-0 = {http://dx.doi.org/10.4208/cicp.240912.180613a},
doi = {10.4208/cicp.240912.180613a},
file = {dijkstra_13_numerical.pdf},
interhash = {51ad5e7f490eaa265d4488da7a5581fe},
intrahash = {02b55afd092d4f47493e60f4ffea7379},
issn = {18152406},
journal = {Communications in Computational Physics},
keywords = {35b32-pdes-bifurcation 35b60-pdes-continuation-and-prolongation-of-solutions 35q35-pdes-in-connection-with-fluid-mechanics 37h20-random-dynamical-systems-bifurcation-theory 37m05-simulation-of-dynamical-systems 65h20-global-methods-including-homotopy-approaches 65p30-numerical-problems-in-dynamical-systems-bifurcation 76-02-fluid-mechanics-research-exposition 76e06-hydrodynamic-stability-convection 76e30-hydrodynamic-stability-nonlinear-effects},
month = jan,
number = 1,
pages = {1 - 45},
posted-at = {2015-11-11 22:14:31},
priority = {2},
timestamp = {2021-01-28T00:30:17.000+0100},
title = {Numerical Bifurcation Methods and their Application to Fluid Dynamics: Analysis beyond Simulation},
url = {https://www.cambridge.org/core/journals/communications-in-computational-physics/article/abs/numerical-bifurcation-methods-and-their-application-to-fluid-dynamics-analysis-beyond-simulation/50CAA5AB617E7A23A44C6E90965F70A1},
volume = 15,
year = 2014
}