Abstract
A collection of methods is presented to adapt a pre-existing timestepping code to perform various bifurcation-theoretic tasks. It is shown that the implicit linear step of a time-stepping code can serve as a highly effective preconditioner for solving linear systems involving the full Jacobian via conjugate gradient iteration. The methods presented for steady-state solving, continuation, direct calculation of bifurcation points (all via Newton’s method), and linear stability analysis (via the inverse power method) rely on this preconditioning. Another set of methods can have as their basis any time-stepping method. These perform various types of stability analyses: linear stability analysis via the exponential power method, Floquet stability analysis of a limit cycle, and nonlinear stability analysis for determining the character of a bifurcation. All of the methods presented require minimal changes to the time-stepping code.
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References
W.E. Arnoldi,The principle of minimized iterations in the solution of the matrix eigenvalue problem, Q. Appl. Math. 9 (1951), 17–29.
D. Barkley and L.S. Tuckerman, Traveling waves in axisymmetric convection: the role of sidewall conductivity, Physica D 37 (1989), 288–294.
D. Barkley and R. Henderson, Floquet stability analysis of the periodic wake of a circular cylinder, J. Fluid Mech. 322 (1996), 215–241.
D. Barkley and L.S. Tuckerman, Stokes preconditioning for the inverse power method, in Lecture Notes in Physics: Proc. of the Fifteenth Int’l. Conf. on Numerical Methods in Fluid Dynamics, P. Kutler, J. Flores, and J.-J. Chattot, eds., Springer, New York, 1997, 75–76.
D. Barkley and L.S. Tuckerman, Stability analysis of perturbed plane Couette flow, Phys. Fluids, submitted (1998).
D. Barkley, M.G.M. Gomes, and R. Henderson, Three-dimensional stability analysis of flow over a backward facing step, J. Fluid Mech., submitted (1998).
A. Bergeon, D. Henry, H. Benhadid, and L.S. Tuckerman, Marangoni convection in binary mixtures with Soret effect, J. Fluid Mech., in press (1998).
F. Bertagnolio, L.S. Tuckerman, P. Le QUéRé, and O. Daube, Calculation of leading eigenmodes in natural convection by the inverse power method with Stokes preconditioning, Scientific report, Laboratoire d’Informatique pour la Mécanique et les Sciences de l’Ingénieur, Orsay, 1998.
E. ChéNIER, Etude de la stabilité linéaire des écoulements thermocapillaries et thermogravitationnels en croissance cristalline Thesis, Université de Paris XI; Notes et Documents LIMSI No. 97–26, 1997.
K.N. Christodoulou and L.E. Scriven, Finding leading modes of a viscous free surface flow: an asymmetric generalized eigenproblem, J. Sci. Comput. 3 (1988), 355–406.
I. Goldhirsch, S.A. Orszag and B.K. Maulik, An efficient method for computing leading eigenvalues and eigenvectors of large asymmetric matrices, J. Sci. Comput. 2 (1987), 33–58.
R.D. Henderson and D. Barkley, Secondary instability in the wake of a circular cylinder, Phys. Fluids 8 (1996), 1683–1685.
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 1995.
K. Lust, Numerical bifurcation analysis of periodic solutions of partial differential equations, PhD thesis, Katholieke Universiteit Leuven, 1997.
C.K. Mamun and L.S. Tuckerman, Asymmetry and Hopf bifurcation in spherical Couette ffow, Phys. of Fluids 7 (1995), 80–91.
P.S. Marcus and L.S. Tuckerman, Numerical simulation of spherical Couette ffow. Part I: Numerical methods and steady states, J. Fluid Mech. 185 (1987), 1–30.
P.S. Marcus and L.S. Tuckerman, Numerical simulation of spherical Couette ffow. Part II: Transitions, J. Fluid Mech. 185 (1987), 31–65.
D.R. Kincaid, T.C. Oppe, and W.D. Joubert, An overview ofNSPCG: A nonsymmetric preconditioned conjugate gradient package, Report CNA-228, Center for Numerical Analysis, University of Texas at Austin, 1988.
Y. Saad,Variations on Arnoldi’s method for computing eigenelements of large unsymmetric matrices, Linear Alg. Appl. 34 (1980), 269–295.
M.F. Schatz, D. Barkley, and H.L. Swinney, Instabilities in spatially periodic channel ffow, Phys. Fluids 7 (1995), 344–358.
R. Seydel, Practical Bifurcation and Stability Analysis, Second edition, Springer, New York, 1994.
R. Touihri Stabilité des écoulements dans une cavité cylindrique chauffée par le bas en présence d’un champ magnétique, Thesis, Ecole Centrale de Lyon, 1998.
L.S. Tuckerman and D. Barkley, Global bifurcation to travelling waves in axisymmetric convection, Phys. Rev. Lett. 61 (1988), 408–411.
L.S. Tuckerman, Steady-state solving via Stokes preconditioning; recursion relations for elliptic operators, in Lecture Notes in Physics: Proc. of the Eleventh Int’l. Conf. on Numerical Methods in Fluid Dynamics, D.L. Dwoyer, M.Y. Hussaini, and R.G. Voigt, eds., Springer, New York, 1989, 573–577.
S. Xin, P. Le QUéRé, and L.S. Tuckerman, Bifurcation analysis of doublediffusive convection with opposing horizontal thermal and solutal gradients Phys. of Fluids. 10 (1998), 850–858.
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Tuckerman, L.S., Barkley, D. (2000). Bifurcation Analysis for Timesteppers. In: Doedel, E., Tuckerman, L.S. (eds) Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems. The IMA Volumes in Mathematics and its Applications, vol 119. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1208-9_20
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DOI: https://doi.org/10.1007/978-1-4612-1208-9_20
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