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Volume 15, Issue 1
Numerical Bifurcation Methods and Their Application to Fluid Dynamics: Analysis Beyond Simulation

Henk A. Dijkstra, Fred W. Wubs, Andrew K. Cliffe, Eusebius Doedel, Ioana F. Dragomirescu, Bruno Eckhardt, Alexander Yu. Gelfgat, Andrew L. Hazel, Valerio Lucarini, Andy G. Salinger, Erik T. Phipps, Juan Sanchez-Umbria, Henk Schuttelaars, Laurette S. Tuckerman & Uwe Thiele

DOI: 10.4208/cicp.240912.180613a

Commun. Comput. Phys., 15 (2014), pp. 1-45.

Published online: 2014-01

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  • 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.

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@Article{CiCP-15-1, author = {A. Dijkstra , HenkW. Wubs , FredK. Cliffe , AndrewDoedel , EusebiusF. Dragomirescu , IoanaEckhardt , BrunoYu. Gelfgat , AlexanderL. Hazel , AndrewLucarini , ValerioG. Salinger , AndyT. Phipps , ErikSanchez-Umbria , JuanSchuttelaars , HenkS. Tuckerman , Laurette and Thiele , Uwe}, title = {Numerical Bifurcation Methods and Their Application to Fluid Dynamics: Analysis Beyond Simulation}, journal = {Communications in Computational Physics}, year = {2014}, volume = {15}, number = {1}, pages = {1--45}, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.240912.180613a}, url = {http://global-sci.org/intro/article_detail/cicp/7086.html} }
TY - JOUR T1 - Numerical Bifurcation Methods and Their Application to Fluid Dynamics: Analysis Beyond Simulation AU - A. Dijkstra , Henk AU - W. Wubs , Fred AU - K. Cliffe , Andrew AU - Doedel , Eusebius AU - F. Dragomirescu , Ioana AU - Eckhardt , Bruno AU - Yu. Gelfgat , Alexander AU - L. Hazel , Andrew AU - Lucarini , Valerio AU - G. Salinger , Andy AU - T. Phipps , Erik AU - Sanchez-Umbria , Juan AU - Schuttelaars , Henk AU - S. Tuckerman , Laurette AU - Thiele , Uwe JO - Communications in Computational Physics VL - 1 SP - 1 EP - 45 PY - 2014 DA - 2014/01 SN - 15 DO - http://doi.org/10.4208/cicp.240912.180613a UR - https://global-sci.org/intro/article_detail/cicp/7086.html KW - AB -

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.

A. Dijkstra , HenkW. Wubs , FredK. Cliffe , AndrewDoedel , EusebiusF. Dragomirescu , IoanaEckhardt , BrunoYu. Gelfgat , AlexanderL. Hazel , AndrewLucarini , ValerioG. Salinger , AndyT. Phipps , ErikSanchez-Umbria , JuanSchuttelaars , HenkS. Tuckerman , Laurette and Thiele , Uwe. (2014). Numerical Bifurcation Methods and Their Application to Fluid Dynamics: Analysis Beyond Simulation. Communications in Computational Physics. 15 (1). 1-45. doi:10.4208/cicp.240912.180613a
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