The numerical analysis of bifurcation problems with application to fluid mechanics

In this review we discuss bifurcation theory in a Banach space setting using the singularity theory developed by Golubitsky and Schaeffer to classify bifurcation points. The numerical analysis of bifurcation problems is discussed and the convergence theory for several important bifurcations is described for both projection and finite difference methods. These results are used to provide a convergence theory for the mixed finite element method applied to the steady incompressible Navier–Stokes equations. Numerical methods for the calculation of several common bifurcations are described and the performance of these methods is illustrated by application to several problems in fluid mechanics. A detailed description of the Taylor–Couette problem is given, and extensive numerical and experimental results are provided for comparison and discussion.

[1]  A. Mallock,et al.  Experiments on Fluid Viscosity , 1896 .

[2]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[3]  L. Rayleigh.  On the Dynamics of Revolving Fluids , 1917 .

[4]  S. Chandrasekhar.  The stability of viscous flow between rotating cylinders , 1954, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[5]  M. Block,et al.  Surface Tension as the Cause of Bénard Cells and Surface Deformation in a Liquid Film , 1956, Nature.

[6]  J. Pearson,et al.  On convection cells induced by surface tension , 1958, Journal of Fluid Mechanics.

[7]  D. Coles.  Transition in circular Couette flow , 1965, Journal of Fluid Mechanics.

[8]  P. Anselone,et al.  AN EXTENSION OF THE NEWTON-KANTOROVIC METHOD FOR SOLVING NONLINEAR EQUATIONS, , 1966 .

[9]  S. S. Antman,et al.  Bifurcation theory and nonlinear eigenvalue problems, 1967 , 1968 .

[10]  M. Crandall,et al.  Bifurcation from simple eigenvalues , 1971 .

[11]  I. Stakgold,et al.  Branching of Solutions of Nonlinear Equations , 1971 .

[12]  E. Riks.  The Application of Newton's Method to the Problem of Elastic Stability , 1972 .

[13]  M. A. Krasnoselʹskii,et al.  Approximate Solution of Operator Equations , 1972 .

[14]  Françoise Chatelin,et al.  Convergence of Approximation Methods to Compute Eigenelements of Linear Operations , 1973 .

[15]  D. Sather,et al.  Branching of solutions of nonlinear equations , 1973 .

[16]  Michael G. Crandall,et al.  Bifurcation, perturbation of simple eigenvalues, itand linearized stability , 1973 .

[17]  D. Sattinger.  Bifurcation at a simple eigenvalue , 1973 .

[18]  D. Sattinger.  Topics in stability and bifurcation theory , 1973 .

[19]  Richard Weiss.  On the Approximation of Fixed Points of Nonlinear Compact Operators , 1974 .

[20]  Edward L. Reiss,et al.  Multiple Eigenvalues Lead to Secondary Bifurcation , 1975 .

[21]  Michael G. Crandall,et al.  The Hopf Bifurcation Theorem in infinite dimensions , 1977 .

[22]  W. Rheinboldt.  Numerical methods for a class of nite dimensional bifur-cation problems , 1978 .

[23]  H. Schwetlick,et al.  Zur Lösung parameterabhängiger nichtlinearer Gleichungen mit singulären Jacobi-Matrizen , 1978 .

[24]  F. Kikuchi.  Finite element approximations to bifurcation problems of turning point type , 1979 .

[25]  R. Seydel.  Numerical computation of branch points in ordinary differential equations , 1979 .

[26]  J. Marsden,et al.  A mathematical introduction to fluid mechanics , 1979 .

[27]  Martin Golubitsky,et al.  Imperfect bifurcation in the presence of symmetry , 1979 .

[28]  Martin Golubitsky,et al.  A Theory for Imperfect Bifurcation via Singularity Theory. , 1979 .

[29]  R. Seydel.  Numerical computation of branch points in nonlinear equations , 1979 .

[30]  W. Langford.  Periodic and Steady-State Mode Interactions Lead to Tori , 1979 .

[31]  T. N. Stevenson,et al.  Fluid Mechanics , 2021, Nature.

[32]  G. Moore,et al.  The numerical treatment of non-trivial bifurcation points , 1980 .

[33]  G. Moore,et al.  The Calculation of Turning Points of Nonlinear Equations , 1980 .

[34]  P. Hall.  Centrifugal instabilities in finite containers: a periodic model , 1980, Journal of Fluid Mechanics.

[35]  David G. Schaeffer,et al.  Qualitative analysis of a model for boundary effects in the Taylor problem , 1980, Mathematical Proceedings of the Cambridge Philosophical Society.

[36]  Jacques Rappaz,et al.  Finite Dimensional Approximation of Non-Linear Problems .1. Branches of Nonsingular Solutions , 1980 .

[37]  M. Yamaguti,et al.  Structure of singularities and its numerical realization in nonlinear elasticity , 1980 .

[38]  F. Thomasset.  Finite element methods for Navier-Stokes equations , 1980 .

[39]  Hans D. Mittelmann,et al.  Bifurcation Problems and their Numerical Solution , 1980 .

[40]  Rita Meyer-Spasche,et al.  Computations of the axisymmetric flow between rotating cylinders , 1980 .

[41]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[42]  B. Mercier,et al.  Eigenvalue approximation by mixed and hybrid methods , 1981 .

[43]  Wolf-Jürgen Beyn,et al.  Stability and Multiplicity of Solutions to Discretizations of Nonlinear Ordinary Differential Equations , 1981 .

[44]  John Guckenheimer,et al.  On a codimension two bifurcation , 1981 .

[45]  F. Brezzi,et al.  Finite dimensional approximation of nonlinear problems , 1981 .

[46]  T. Mullin,et al.  Anomalous modes in the Taylor experiment , 1981, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[47]  H. Swinney,et al.  Instabilities and transition in flow between concentric rotating cylinders , 1981 .

[48]  Computation of transitions in Taylor vortex flows , 1981 .

[49]  F. Brezzi,et al.  Finite dimensional approximation of nonlinear problems , 1981 .

[50]  Robert L. Lee,et al.  The cause and cure (!) of the spurious pressures generated by certain fem solutions of the incompressible Navier‐Stokes equations: Part 2 , 1981 .

[51]  Approximation of Hopf bifurcation , 1982 .

[52]  P. Hall.  Centrifugal instabilities of circumferential flows in finite cylinders: the wide gap problem , 1982, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[53]  André Vanderbauwhede,et al.  Local bifurcation and symmetry , 1982 .

[54]  T. Brooke Benjamin,et al.  Notes on the multiplicity of flows in the Taylor experiment , 1982, Journal of Fluid Mechanics.

[55]  Alastair Spence,et al.  Non-simple Turning Points and Cusps , 1982 .

[56]  J. Descloux,et al.  Approximation of Solution Branches of Non-Linear Equations , 1982 .

[57]  K. A. Cliffe,et al.  Numerical calculations of two-cell and single-cell Taylor flows , 1983, Journal of Fluid Mechanics.

[58]  Milan Kubíček,et al.  Book-Review - Computational Methods in Bifurcation Theory and Dissipative Structures , 1983 .

[59]  W. Rheinboldt,et al.  On the Discretization Error of Parametrized Nonlinear Equations , 1983 .

[60]  Andreas Griewank,et al.  The Calculation of Hopf Points by a Direct Method , 1983 .

[61]  A. Spence,et al.  THE COMPUTATION OF SYMMETRY-BREAKING BIFURCATION POINTS* , 1984 .

[62]  Approximation of Hopf Bifurcation for Semilinear Parabolic Equations , 1984 .

[63]  W. Rheinboldt,et al.  Solution manifolds and submanifolds of parametrized equations and their discretization errors , 1984 .

[64]  A. Spence,et al.  The Calculation of High Order Singularities in the Finite Taylor Problem , 1984 .

[65]  Allan D. Jepson,et al.  Singular Points and their Computation , 1984 .

[66]  A. Griewank,et al.  Characterization and Computation of Generalized Turning Points , 1984 .

[67]  Josef Hofbauer,et al.  A Hopf bifurcation theorem for difference equations approximating a differential equation , 1984 .

[68]  Wolf-Jürgen Beyn,et al.  Defining Equations for Singular Solutions and Numerical Applications , 1984 .

[69]  H. Mittelmann,et al.  Numerical methods for bifurcation problems : proceedings of the conference at the University of Dortmund, August 22-26, 1983 , 1984 .

[70]  Mullin.  Onset of time dependence in Taylor-Couette flow. , 1985, Physical review. A, General physics.

[71]  Allan D. Jepson,et al.  Folds in Solutions of Two Parameter Systems and Their Calculation. Part I , 1985 .

[72]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[73]  T. Mullin,et al.  A numerical and experimental study of anomalous modes in the Taylor experiment , 1985, Journal of Fluid Mechanics.

[74]  Finite element solutions of the Navier-Stokes equations , 1985 .

[75]  W. Rheinboldt,et al.  LOCAL ERROR ESTIMATES FOR PARAMETRIZED NONLINEAR EQUATIONS , 1985 .

[76]  Allan D. Jepson,et al.  The Numerical Solution of Nonlinear Equations Having Several Parameters I: Scalar Equations , 1985 .

[77]  H. Weinitschke.  On the calculation of limit and bifurcation points in stability problems of elastic shells , 1985 .

[78]  Rita Meyer-Spasche,et al.  Some Bifurcation Diagrams for Taylor Vortex Flows , 1985 .

[79]  Dirk Roose,et al.  A Direct Method for the Computation of Hopf Bifurcation Points , 1985 .

[80]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[81]  Operator approximation and symmetry-breaking bifurcation , 1986 .

[82]  A. Bossavit,et al.  Symmetry, groups and boundary value problems. A progressive introduction to noncommutative harmonic analysis of partial differential equations in domains with geometrical symmetry , 1986 .

[83]  J. T. Stuart,et al.  Taylor-vortex flow: a dynamical system , 1986 .

[84]  M. Nagata,et al.  Bifurcations in Couette flow between almost corotating cylinders , 1986, Journal of Fluid Mechanics.

[85]  Harry L. Swinney,et al.  Flow regimes in a circular Couette system with independently rotating cylinders , 1986, Journal of Fluid Mechanics.

[86]  G. Iooss.  Secondary bifurcations of Taylor vortices into wavy inflow or outflow boundaries , 1986, Journal of Fluid Mechanics.

[87]  W. Rheinboldt.  Numerical analysis of parametrized nonlinear equations , 1986 .

[88]  K. Cliffe,et al.  The use of symmetry in bifurcation calculations and its application to the Be´nard problem , 1986 .

[89]  Numerical analysis of bifurcation problems of nonlinear equations , 1986 .

[90]  Ian Stewart,et al.  Symmetry and stability in Taylor Couette flow , 1986 .

[91]  G. Schrauf.  The first instability in spherical Taylor-Couette flow , 1986, Journal of Fluid Mechanics.

[92]  Hans D Mittlemann.  A pseudo-arclength continuation method for nonlinear eigenvalue problems , 1986 .

[93]  Laurette S. Tuckerman,et al.  Simulation of flow between concentric rotating spheres. Part 1. Steady states , 1987, Journal of Fluid Mechanics.

[94]  K. H. Winters,et al.  A bifurcation study of laminar flow in a curved tube of rectangular cross-section , 1987, Journal of Fluid Mechanics.

[95]  H. B. Keller,et al.  Computation of anomalous modes in the Taylor experiment , 1987 .

[96]  Willy Govaerts,et al.  Numerical methods for bifurcations of dynamical equilibria , 1987 .

[97]  Gerald Moore.  SOME REMARKS ON THE DEFLATED BLOCK ELIMINATION METHOD , 1987 .

[98]  Pfister,et al.  Unusual time-dependent phenomena in Taylor-Couette flow at moderately low Reynolds numbers. , 1987, Physical review letters.

[99]  C. P. Jackson.  A finite-element study of the onset of vortex shedding in flow past variously shaped bodies , 1987, Journal of Fluid Mechanics.

[100]  L. E. Scriven,et al.  Finding leading modes of a viscous free surface flow: An asymmetric generalized eigenproblem , 1988, J. Sci. Comput..

[101]  Gerd Pfister,et al.  Bifurcation phenomena in Taylor-Couette flow in a very short annulus , 1988, Journal of Fluid Mechanics.

[102]  Vladimir Janovsky.  Minimally extended defining conditions for singularities of codim≤2 , 1988 .

[103]  K. Cliffe.  Numerical calculations of the primary-flow exchange process in the Taylor problem , 1988, Journal of Fluid Mechanics.

[104]  J. M. Sanz-Serna,et al.  Stability and Convergence in Numerical Analysis III: Linear Investigation of Nonlinear Stability , 1988 .

[105]  K. Cliffe,et al.  The onset of convection in a finite container due to surface tension and buoyancy , 1988 .

[106]  M. Wagner,et al.  Interactions of Secondary Branches of Taylor Vortex Solutions , 1989 .

[107]  A. Spence,et al.  A note on the calculation of paths of Hopf bifurcations , 1989 .

[108]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[109]  A. Griewank,et al.  Computation of cusp singularities for operator equations and their discretizations , 1989 .

[110]  T. Mullin,et al.  Nonlinear flow phenomena in a symmetric sudden expansion , 1990, Journal of Fluid Mechanics.

[111]  H. B. Keller,et al.  Wavy Taylor-vortex flows via multigrid-continuation methods , 1990 .

[112]  T. Mullin,et al.  Novel bifurcation phenomena in a rotating annulus , 1991, Journal of Fluid Mechanics.

[113]  Willy Govaerts,et al.  Stable solvers and block elimination for bordered systems , 1991 .

[114]  F. W. Schneider,et al.  Bifurcation and Chaos: Analysis, Algorithms, Applications , 1991 .

[115]  A. Spence,et al.  Two Methods for the Numerical Detection of Hopf Bifurcations , 1991 .

[116]  W. Beyn.  Numerical methods for dynamical systems , 1991 .

[117]  Bodo Werner,et al.  COMPUTATION OF HOPF BRANCHES BIFURCATING FROM TAKENS-BOGDANOV POINTS FOR PROBLEMS WITH SYMMETRIES , 1991 .

[118]  F. Durst,et al.  Flow through rotating rectangular ducts , 1991 .

[119]  K. Nandakumar,et al.  A bifurcation study of mixed-convection heat transfer in horizontal ducts , 1991, Journal of Fluid Mechanics.

[120]  Primary Flow Exchange Mechanisms in the Taylor Apparatus ApplyingImpermeable Stress-free Boundary Conditions , 1991 .

[121]  K. A. Cliffe,et al.  The role of anomalous modes in Taylor─Couette flow , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[122]  Vladimír Janovský,et al.  Computer-aided analysis of imperfect bifurcation diagrams, I. simple bifurcation point and isola formation centre. , 1992 .

[123]  Krishnaswamy Nandakumar,et al.  An experimental and numerical study of the Dean problem: flow development towards two-dimensional multiple solutions , 1992, Journal of Fluid Mechanics.

[124]  H. Dijkstra.  On the structure of cellular solutions in Rayleigh–Bénard–Marangoni flows in small-aspect-ratio containers , 1992, Journal of Fluid Mechanics.

[125]  Eugene L. Allgower,et al.  Continuation and path following , 1993, Acta Numerica.

[126]  Anne E. Trefethen,et al.  Hydrodynamic Stability Without Eigenvalues , 1993, Science.

[127]  Gautam M. Shroff,et al.  Stabilization of unstable procedures: the recursive projection method , 1993 .

[128]  M. Mcpherson,et al.  Introduction to fluid mechanics , 1997 .

[129]  J. Guckenheimer,et al.  Computing Hopf Bifurcations I , 1997 .

[130]  K. A. Cliffe,et al.  Eigenvalues of the discretized Navier-Stokes equation with application to the detection of Hopf bifurcations , 1993, Adv. Comput. Math..

[131]  A. Spence,et al.  Is the steady viscous incompressible two‐dimensional flow over a backward‐facing step at Re = 800 stable? , 1993 .

[132]  R. Neubert.  PREDICTOR-CORRECTOR TECHNIQUES FOR DETECTING HOPF BIFURCATION POINTS , 1993 .

[133]  Harald Von Sosen.  Part I: Folds and bifurcations in the solutions of semi-explicit differential-algebraic equations. Part II: The recursive projection method applied to differential-algebraic equations and incompressible fluid mechanics , 1994 .

[134]  A. Spence,et al.  Shift-invert and Cayley transforms for detection of rightmost eigenvalues of nonsymmetric matrices , 1994 .

[135]  R. Seydel.  Practical bifurcation and stability analysis : from equilibrium to chaos , 1994 .

[136]  K. Nandakumar,et al.  A bifurcation study of viscous flow through a rotating curved duct , 1994, Journal of Fluid Mechanics.

[137]  L. Tuckerman,et al.  Asymmetry and Hopf bifurcation in spherical Couette flow , 1995 .

[138]  Simon Tavener,et al.  Bifurcation for flow past a cylinder between parallel planes , 1995, Journal of Fluid Mechanics.

[139]  W. Govaerts.  BORDERED MATRICES AND SINGULARITIES OF LARGE NONLINEAR SYSTEMS , 1995 .

[140]  A. Ploeg,et al.  An efficient code to compute non-parallel steady flows and their linear stability , 1995 .

[141]  Willy Govaerts,et al.  Detection of Hopf points by counting sectors in the complex plane , 1996 .

[142]  Andreas Griewank,et al.  The Approximate Solution of Defining Equations for Generalized Turning Points , 1996 .

[143]  Ruey-Jen Yang.  A NUMERICAL PROCEDURE FOR PREDICTING MULTIPLE SOLUTIONS OF A SPHERICAL TAYLOR–COUETTE FLOW , 1996 .

[144]  H. Dijkstra,et al.  A bifurcation study of double diffusive flows in a laterally heated stably stratified liquid layer , 1996 .

[145]  J. Hale,et al.  Methods of Bifurcation Theory , 1996 .

[146]  F. Durst,et al.  Further contributions on the two-dimensional flow in a sudden expansion , 1997, Journal of Fluid Mechanics.

[147]  W. Govaerts.  Computation of Singularities in Large Nonlinear Systems , 1997 .

[148]  D. Drikakis.  Bifurcation phenomena in incompressible sudden expansion flows , 1997 .

[149]  A. Spence,et al.  Hopf bifurcation near a double singular point , 1997 .

[150]  Lloyd N. Trefethen,et al.  Pseudospectra of Linear Operators , 1997, SIAM Rev..

[151]  F. Battaglia,et al.  Bifurcation of Low Reynolds Number Flows in Symmetric Channels , 1996 .

[152]  Karl Meerbergen,et al.  Implicitly restarted Arnoldi with purification for the shift-invert transformation , 1997, Math. Comput..

[153]  K. Meerbergen,et al.  The Restarted Arnoldi Method Applied to Iterative Linear System Solvers for the Computation of Rightmost Eigenvalues , 1997 .

[154]  H. Dijkstra,et al.  Symmetry breaking and overturning oscillations in thermohaline-driven flows , 1997, Journal of Fluid Mechanics.

[155]  Bryan D. Davidson.  Large-Scale Continuation and Numerical Bifurcation for Partial Differential Equations , 1997 .

[156]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[157]  Alexander Ostermann,et al.  Hopf bifurcation of reaction-diffusion and Navier-Stokes equations under discretization , 1998, Numerische Mathematik.

[158]  Dirk Roose,et al.  An Adaptive Newton-Picard Algorithm with Subspace Iteration for Computing Periodic Solutions , 1998, SIAM J. Sci. Comput..

[159]  Karl Meerbergen,et al.  Using Generalized Cayley Transformations within an Inexact Rational Krylov Sequence Method , 1998, SIAM J. Matrix Anal. Appl..

[160]  Andrew Knyazev,et al.  Preconditioned Eigensolvers - an Oxymoron? , 1998 .

[161]  R. Sani,et al.  Incompressible Flow and the Finite Element Method, Volume 1, Advection-Diffusion and Isothermal Laminar Flow , 1998 .

[162]  Bifurcation analysis of double-diffusive convection with opposing horizontal thermal and solutal gradients , 1998 .

[163]  L. Tuckerman,et al.  Marangoni convection in binary mixtures with Soret effect , 1998, Journal of Fluid Mechanics.

[164]  Simon Tavener,et al.  Marangoni-Bénard Convection with a Deformable Free Surface , 1998 .

[165]  P. Love,et al.  Bifurcations in Kolmogorov and Taylor-vortex flows , 1999 .

[166]  K. A. Cliffe,et al.  Implementation of extended systems using symbolic algebra , 2000 .

[167]  Simon Tavener,et al.  International Journal for Numerical Methods in Fluids O(2)-symmetry Breaking Bifurcation: with Application to the Flow past a Sphere in a Pipe , 2022 .

[168]  J. Szmelter.  Incompressible flow and the finite element method , 2001 .

[169]  Richard B. Lehoucq,et al.  Large‐scale eigenvalue calculations for stability analysis of steady flows on massively parallel computers , 2001 .