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Numerical Bifurcation Methods and their Application to Fluid Dynamics: Analysis beyond Simulation

Published online by Cambridge University Press:  03 June 2015

Henk A. Dijkstra*
Institute for Marine and Atmospheric Research Utrecht, Utrecht University, The Netherlands
Fred W. Wubs
Department of Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands
Andrew K. Cliffe
School of Mathematical Sciences, University of Nottingham, Nottingham, UK
Eusebius Doedel
Department of Computer Science, Concordia University, Montreal, Canada
Ioana F. Dragomirescu
National Centre for Engineering Systems of Complex Fluids, University Politehnica of Timisoara, Romania
Bruno Eckhardt
Fachbereich Physik, Philipps-Universität Marburg, Marburg, Germany
Alexander Yu. Gelfgat
School of Mechanical Engineering, Faculty of Engineering, Tel-Aviv University, Tel-Aviv, Israel
Andrew L. Hazel
School of Mathematics, University of Manchester, Manchester, UK
Valerio Lucarini
Meteorological Institute, Klimacampus, University of Hamburg, Hamburg, Germany Department of Mathematics and Statistics, University of Reading, Reading, UK
Andy G. Salinger
Sandia National Laboratories, Albuquerque, USA
Erik T. Phipps
Sandia National Laboratories, Albuquerque, USA
Juan Sanchez-Umbria
Departament de Fisica Aplicada, Universitat Politecnica de Catalunya, Barcelona, Spain
Henk Schuttelaars
Department of Applied Mathematical Analysis, TU Delft, Delft, the Netherlands
Laurette S. Tuckerman
PMMH-ESPCI, Paris, France
Uwe Thiele
Department of Mathematical Sciences, Loughborough University, Loughborough, UK
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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 arementioned 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.

Review Article
Copyright © Global Science Press Limited 2014

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