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 Gelfgat, Andrew L. Hazel, Valerio Lucarini, Andy G. Salinger, Erik T. Phipps, Sanchez Umbria Juan, Henk Schuttelaars, Laurette S. Tuckerman, Uwe Thiele

Research output: Contribution to journalReview articlepeer-review

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 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.

Original languageEnglish
Pages (from-to)1-45
Number of pages45
JournalCommunications in Computational Physics
Volume15
Issue number1
DOIs
StatePublished - Jan 2014

Keywords

  • High-dimensional dynamical systems
  • Numerical bifurcation analysis
  • Transitions in fluid flows

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