Hostname: page-component-66d9dcfd78-4fl2k Total loading time: 0 Render date: 2026-08-08T04:39:59.067Z Has data issue: false hasContentIssue false
  • English
  • Français

Laplacian Preconditioning for the Inverse Arnoldi Method

Part of: Approximation methods and numerical treatment of dynamical systems Numerical problems in dynamical systems Numerical linear algebra Hydrodynamic stability

Published online by Cambridge University Press:  23 November 2015

Laurette S. Tuckerman
Laurette S. Tuckerman*
Affiliation:
PMMH(UMR 7636 CNRS - ESPCI - UPMC Paris 6 - UPD Paris 7 - ParisTech - PSL), 10 rue Vauquelin, 75005 Paris, France
*
*Corresponding author. Email address:laurette@pmmh.espci.fr(L. S. Tuckerman)
Get access

Abstract

Many physical processes are described by elliptic or parabolic partial differential equations. For linear stability problems associated with such equations, the inverse Laplacian provides a very effective preconditioner. In addition, it is also readily available in most scientific calculations in the form of a Poisson solver or an implicit diffusive timestep. We incorporate Laplacian preconditioning into the inverse Arnoldi method, using BiCGSTAB to solve the large linear systems. Two successful implementations are described: spherical Couette flow described by the Navier-Stokes equations and Bose-Einstein condensation described by the nonlinear Schrödinger equation.

Keywords

To be provided by author

MSC classification

Secondary: 37M20: Computational methods for bifurcation problems 65F08: Preconditioners for iterative methods 65F18: Inverse eigenvalue problems 65P10: Hamiltonian systems including symplectic integrators 65P30: Bifurcation problems 76E07: Rotation

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 5 , November 2015 , pp. 1336 - 1351
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Arnoldi, W.E., The principle of minimized iterations in the solution of the matrix eigenvalue problem, Q. Appl. Math 9, 17 (1951).Google Scholar
[2]Tuckerman, L.S., Steady state solving via Stokes preconditioning; recursion relations for elliptic operators, in Lecture Notes in Physics (No. 323): Proc. of the 11th Int'l. Conf. on Numerical Methods in Fluid Dynamics, ed. by Dwoyer, D.L., Hussaini, M.Y. & Voigt, R.G., (Springer, New York, 1989), p. 573–577.Google Scholar
[3]Mamun, C.K. and Tuckerman, L.S., Asymmetry and Hopfbifurcation in spherical Couette flow, Phys. Fluids 7,80 (1995).CrossRefGoogle Scholar
[4]Bergeon, A., Henry, D., Benhadid, H. and Tuckerman, L.S., Marangoni convection in binary mixtures with Soret effect, J. Fluid Mech. 375,143 (1998).CrossRefGoogle Scholar
[5]Huepe, C., Métens, S., Dewel, G., Borckmans, P. and Brachet, M.E., Decay rates in attractive Bose-Einstein condensates, Phys. Rev. Lett. 82,1616 (1999).CrossRefGoogle Scholar
[6]Batiste, O., Knobloch, E., Alonso, A. and Mercader, I., J. Fluid Mech. 560,149 (2006).CrossRefGoogle Scholar
[7]Tuckerman, L.S. and Barkley, D., Bifurcation analysis for time-steppers, in Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, ed. by Doedel, E. and Tuckerman, L.S. (Springer, New York, 2000), p. 452–466.Google Scholar
[8]Barkley, D. and Tuckerman, L.S., Stokes preconditioning for the inverse power method, in Lecture Notes in Physics: Proc. of the 15th Int’l. Conf. on Numerical Methods in Fluid Dynamics ed. by Kutler, P., Flores, J. and Chattot, J.-J. (Springer, New York, 1997), p. 75–76.Google Scholar
[9]Tuckerman, L.S., Huepe, C. and Brachet, M.-E., Numerical methods for bifurcation problems, in Instabilities and non-equilibrium structures IX, ed. by Descalzi, O., Martinez, J. and Rica, S. (Kluwer, Dordecht, 2004).Google Scholar
[10]Huepe, C., Tuckerman, L.S., Métens, S., and Brachet, M.E., Stability and decay rates of non-isotropic attractive Bose-Einstein condensates, Phys. Rev. A. 68, 023609 (2003).Google Scholar
[11]van der Vorst, H.A., Bi-CGSTAB: A fast and smoothly converging variant ofBi-CGfor the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 13,631 (1992).CrossRefGoogle Scholar
[12]Gottlieb, D. and Orszag, S. A., Numerical Analysis of Spectral Methods(SIAM, Philadelphia, 1977).Google Scholar
[13]Boyd, J.P., Chebyshev and Fourier Spectral Methods (Dover, New York, 2001).Google Scholar
[14]Wimmer, M., Experiments on a viscous fluid flow between concentric rotating spheres, J. Fluid Mech. 78,317 (1976).Google Scholar
[15]Schrauf, G., Branching of Navier-Stokes equations in a spherical gap, in Lecture Notes in Physics: Proc. of the 8th Int’l. Conf. on Numerical Methods in Fluid Dynamics, ed. by Krause, E. (Springer, New York, 1982), p. 474.Google Scholar
[16]Tuckerman, L.S. & Marcus, P.S., Formation of Taylor vortices in spherical Couette flow, in Lecture Notes in Physics (No. 218): Proc. of the 9th Int'l. Conf. on Numerical Methods in Fluid Dynamics, ed. by Soubbarameyer, & Boujot, J.P., (Springer, New York, 1985), p. 552–556.Google Scholar
[17]Schrauf, G., The first instability in spherical Taylor-Couetteflow, J. Fluid Mech. 166,287 (1986).CrossRefGoogle Scholar
[18]Marcus, P.S. and Tuckerman, L.S., Numerical simulation of spherical Couette flow. Part I: Numerical methods and steady states, J. Fluid Mech. 185,1 (1987).CrossRefGoogle Scholar
[19]Marcus, P.S. and Tuckerman, L.S., Numerical simulation of spherical Couette flow. Part II: Transitions, J. Fluid Mech. 185,31 (1987).Google Scholar
[20]Gross, E.P., Nuovo Cimento 20 454 (1961).Google Scholar
[21]Pitaevskii, L.P., Vortex lines in an imperfect Bose gas, Zh. Eksp. Teor. Fiz. 40, 646 (1961) [Sov. Phys. JETP 13,451 (1961)].Google Scholar
[22]Bose, S., Plancks Gesetz und Lichtquantenhypothese, Z. Phys. 26,178 (1924).Google Scholar
[23]Einstein, A., Quantentheorie des einatomigen idealen gases: Zweite Abhandlung, Sitzungber. Preuss. Akad. Wiss. 1925,3 (1925).Google Scholar
[24]Anderson, M.H., Ensher, J.R., Matthews, M.R., Wieman, C.E. and Cornell, E.A., Observation of Bose-Einstein condensation in a dilute atomic vapor, Science 269,198 (1995).Google Scholar
[25]Davis, K.B., Mewes, M.-O., Andrews, M.R., van Druten, N.J., Durfee, D.S., Kurn, D.M. and Ketterle, W., Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett. 75,3969 (1995).Google Scholar
[26]Bradley, C.C., Sackett, C.A., Tollett, J.J. and Hulet, R.G., Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions, Phys. Rev. Lett. 75,1687 (1995).Google Scholar
[27]Ruprecht, P.A., Holland, M.J., Burnett, K., and Edwards, M., Time-dependent solution of the nonlinear Schrb'dinger equation for Bose-condensed trapped neutral atoms, Phys. Rev. A 51, 4704 (1995).Google Scholar
[28]Ueda, M. and Leggett, A.J., Macroscopic quantum tunneling of a Bose-Einstein condensate with attractive interaction, Phys. Rev. Lett. 80,1576 (1998).CrossRefGoogle Scholar
[29]Roberts, J.L., Claussen, N.R., Cornish, S.L., Donley, E.A., Cornell, E.A. and Wieman, C.E., Controlled collapse of a Bose-Einstein condensate, Phys. Rev. Lett. 86,4211 (2001).Google Scholar
[30]Gammal, A., Frederico, T. and Tomio, L., Critical number of atoms for attractive Bose-Einstein condensates with cylindrically symmetrical traps, Phys. Rev. A 64 055602 (2001).Google Scholar