Abstract
With the ever increasing computational power available and the development of high-performances computing, investigating the properties of realistic very large-scale nonlinear dynamical systems has become reachable. It must be noted however that the memory capabilities of computers increase at a slower rate than their computational capabilities. Consequently, the traditional matrix-forming approaches wherein the Jacobian matrix of the system considered is explicitly assembled become rapidly intractable. Over the past two decades, so-called matrix-free approaches have emerged as an efficient alternative. The aim of this chapter is thus to provide an overview of well-grounded matrix-free methods for fixed points computations and linear stability analyses of very large-scale nonlinear dynamical systems.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Note that the non-normality of \(\varvec{\mathcal {A}}\) also implies that its right and left eigenvectors are different. This observation may have large consequences in fluid dynamics, particularly when addressing the problems of optimal linear control and/or estimation of strongly non-parallel flows.
- 2.
Formally, a convex optimization problem reads
$$\begin{aligned} \begin{aligned} \mathop {\text {manimize}}\limits _{\mathbf {x}}\,&\mathcal {J} \left( \mathbf {x} \right) \\ \mathop {\text {subject~to}}&\, g_i \left( \mathbf {x} \right) \le 0, \ i = 1, \ldots , m \\ ~&h_i \left( \mathbf {x} \right) = 0, \ i = 1, \ldots , p, \end{aligned} \end{aligned}$$where the objective function \(\mathcal {J} \left( \mathbf {x} \right) \) and the inequality constraints functions \(g_i \left( \mathbf {x} \right) \) are convex. The conditions on the equality constraints functions \(h_i \left( \mathbf {x} \right) \) are more restrictive as they need to be affine functions, i.e. of the form \(h_i \left( \mathbf {x} \right) = \mathbf {a}_i^T \mathbf {x} + b_i\). See the book by Boyd and Vandenberghe [9] for extensive details about convex optimization.
- 3.
Given an appropriate inner product, the adjoint operator \(\varvec{\mathcal {A}}^{\dagger }\) is defined such that
$$\begin{aligned} \langle \mathbf {v} \vert \varvec{\mathcal {A}} \mathbf {x} \rangle = \langle \varvec{\mathcal {A}}^{\dagger } \mathbf {v} \vert \mathbf {x} \rangle , \end{aligned}$$where \(\langle \mathbf {a} \vert \mathbf {b} \rangle \) denotes the inner product of \(\mathbf {a}\) and \(\mathbf {b}\). If one consider the classical Euclidean inner product, the adjoint operator is simply given by
$$\varvec{\mathcal {A}}^{\dagger } = \varvec{\mathcal {A}}^H$$where \(\varvec{\mathcal {A}}^H\) is the Hermitian (i.e. complex-conjugate transpose) of \(\varvec{\mathcal {A}}\). It must be noted finally that the direct operator \(\varvec{\mathcal {A}}\) and the adjoint one \(\varvec{\mathcal {A}}^{\dagger }\) have the same eigenspectrum. This last observation is a key point when one aims at validating the numerical implementation of an adjoint solver.
References
Intel Math Kernel Library. Reference Manual. Intel Corporation, 2009
Åkervik, E., Brandt, L., Henningson, D.S., Hœpffner, J., Marxen, O., Schlatter, P.: Steady solutions of the Navier-Stokes equations by selective frequency damping. Phys. Fluids 18(6), 068102 (2006)
Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Ostrouchov, S., Sorensen, D.: Lapack Users’ Guide, 3rd edn. SIAM (1999)
Antoulas, A.C.: Approximation of Large-scale Dynamical Systems. SIAM, Philadelphia (2005)
Arnoldi, W.E.: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q. Appl. Math. 9, 1951 (1951)
Bagheri, S., Åkervik, E., Brandt, L., Henningson, D.S.: Matrix-free methods for the stability and control of boundary layers. AIAA J. 47(5), 1057–1068 (2009)
Bagheri, S., Henningson, D.S., Hoepffner, J., Schmid, P.J.: Input-output analysis and control design applied to a linear model of spatially developing flows. Appl. Mech. Rev. 62(2), 020803 (2009)
Basley, J., Pastur, L.R., Lusseyran, F., Faure, T.M., Delprat, N.: Experimental investigation of global structures in an incompressible cavity flow using time-resolved PIV. Exp. Fluids 50, 905–918 (2011)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Brandt, L.: The lift-up effect: The linear mechanism behind transition and turbulence in shear flows. Eur. J. Mech.-B/Fluids 47, 80–96 (2014)
Bridges, T.J., Morris, P.J.: Differential eigenvalue problems in which the parameter appears nonlinearly. J. Comput. Phys. 55, 437–460 (1984)
Bridges, T.J., Morris, P.J.: Boundary layer stability calculations. Phys. Fluids 30:(11) (1987)
Bucci, A.M.: Subcritical and supercritical dynamics of incompressible flow over miniaturized roughness elements. Ph.D. thesis, École Nationale Supérieure d’Arts et Métiers-ENSAM (2017)
Bucci, M.A., Puckert, D.K., Andriano, C., Loiseau, J.-Ch., Cherubini, S., Robinet, J.-Ch., Rist. U.: Roughness-induced transition by quasi-resonance of a varicose global mode. J. Fluid Mech. 836, 167–191 (2018)
Campobasso, M.S., Giles, M.B.: Stabilization of a linear flow solver for turbomachinery aeroelasticity using recursive projection method. AIAA J. 42(9), 1765–1774 (2004)
Chen, K.K., Rowley, C.W.:. H2 optimal actuator and sensor placement in the linearised complex Ginzburg-Landau system. J. Fluid Mech. 681, 241–260 (2011)
Citro, V., Giannetti, F., Luchini, P., Auteri, F.: Global stability and sensitivity analysis of boundary-layer flows past a hemispherical roughness element. Phys. Fluids 27(8), 084110 (2015)
Citro, V., Luchini, P., Giannetti, F., Auteri, F.: Efficient stabilization and acceleration of numerical simulation of fluid flows by residual recombination. J. Comput. Phys. 344, 234–246 (2017)
Cunha, G., Passaggia, P.-Y., Lazareff, M.: Optimization of the selective frequency damping parameters using model reduction. Phys. Fluids 27(9), 094103 (2015)
Davey, A., Drazin, P.J.: The stability of Poiseuille flow in a pipe. J. Fluid Mech. 36, 209–218 (1969)
Dijkstra, Henk A., Wubs, Fred W., Cliffe, Andrew K., Doedel, Eusebius, Dragomirescu, Ioana F., Eckhardt, Bruno, Gelfgat, Alexander Yu., Hazel, Andrew L., Lucarini, Valerio, Salinger, Andy G., et al.: Numerical bifurcation methods and their application to fluid dynamics: analysis beyond simulation. Commun. Comput. Phys. 15(1), 1–45 (2014)
Edwards, W.S., Tuckerman, L.S., Friesner, R.A., Sorensen, D.C.: Krylov methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 110(1), 82–102 (1994)
Farano, Mirko, Cherubini, Stefania, Robinet, Jean-Christophe, De Palma, Pietro: Subcritical transition scenarios via linear and nonlinear localized optimal perturbations in plane poiseuille flow. Fluid Dyn. Res. 48(6), 061409 (2016)
Faure, T.M., Adrianos, P., Lusseyran, F., Pastur, L.: Visualizations of the flow inside an open cavity at medium range Reynolds numbers. Exp. Fluids 42, 169–184 (2007)
Faure, T.M., Pastur, L., Lusseyran, F., Fraigneau, Y., Bisch, D.: Three-dimensional centrifugal instabilities development inside a parallelepipedic open cavity of various shape. Exp. Fluids 47, 395–410 (2009)
Fischer, P., Kruse, J., Mullen, J., Tufo, H., Lottes, J., Kerkemeier, S.: Open source spectral element CFD solver (2008). https://nek5000.mcs.anl.gov/index.php/MainPage
Foures, D.P.G., Caulfield, C.P., Schmid, P.J.: Optimal mixing in two-dimensional plane Poiseuille flow at finite Péclet number. J. Fluid Mech. 748, 241–277 (2012)
Foures, D.P.G., Caulfield, C.P., Schmid, P.J.: Localization of flow structures using \(\infty \)-norm optimization. J. Fluid Mech. 729, 672–701 (2013)
Foures, D.P.G., Caulfield, C.P., Schmid, P.J.: Variational framework for flow optimization using seminorm constraints. Phys. Rev. E 86(2), 026306 (2014)
Gary, J., Helgason, R.: A matrix method for ordinary differential eigenvalue problems. J. Comput. Phys. 5, 169–187 (1970)
Gaster, M., Jordinson, R.: On the eigenvalues of the Orr-Sommerfeld equation. J. Fluid Mech. 72, 121–133 (1975)
Golub, G.H., Van Loan, C.F.: Matrix Computations, vol. 3. JHU Press (2012)
Hascoët, L., Pascual, V.: The Tapenade Automatic Differentiation tool: principles, model, and specification. ACM Trans. Math. Softw. 39(3), 1–43 (2013)
Hecht, F.: New development in freefem++. J. Numer. Math. 20(3–4), 251–265 (2012)
Hestenes, M.R., Stiefel, E.: Methods of Conjugate Gradients for Solving Linear Systems, vol. 49. NBS Washington, DC (1952)
Hunt, R.E., Crighton, D.G.: Instability of Flows in Spatially Developing Media. In: Proceedings of the Royal Society of London A, vol. 435, pp. 109–128. The Royal Society (1991)
Ilak, M., Schlatter, P., Bagheri, S., Henningson, D.S.: Bifurcation and stability analysis of a jet in cross-flow: onset of global instability at a low velocity ratio. J. Fluid Mech. 696, 94–121 (2012)
Janovskỳ, V., Liberda, O.: Continuation of invariant subspaces via the recursive projection method. Appl. Math. 48(4), 241–255 (2003)
Jordi, B.E., Cotter, C.J., Sherwin, S.J.: Encapsulated formulation of the selective frequency damping method. Phys. Fluids 26(3), 034101 (2014)
Jordi, B.E., Cotter, C.J., Sherwin, S.J.: An adaptive selective frequency damping method. Phys. Fluids 27(9), 094104 (2015)
Jordinson, R.: The flat plate boundary layer. Part 1. Numerical integration of the Orr-Sommerfeld equation. J. Fluid Mech. 43, 801–811 (1970)
Jordinson, R.: Spectrum of eigenvalues of the Orr-Sommerfeld equation for blasius flow. Phys. Fluids 14, 2535 (1971)
Karniadakis, G., Sherwin, S.: Spectral/HP Element Methods for Computational Fluid Dynamics, 2nd edn. Oxford University Press, Oxford (2005)
Kerswell, R.R., Pringle, C.C.T., Willis, A.P.: An optimization approach for analysing nonlinear stability with transition to turbulence in fluids as an exemplar. Rep. Prog. Phys. 77(8), 085901 (2014)
Kerswell, R.R.: Nonlinear nonmodal stability theory. Ann. Rev. Fluid Mech. 50, 319–345 (2018)
Knoll, D.A., Keyes, D.E.: Jacobian-free Newton-Krylov methods: a survey of approaches and applications. J. Comput. Phys. 193(2), 357–397 (2004)
Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Nat. Bur. Stand. Sec. B 45, 255–282 (1950)
Landahl, M.T.: A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98(2), 243–251 (1980)
Loiseau, Jean-Christophe, Robinet, Jean-Christophe, Cherubini, Stefania, Leriche, Emmanuel: Investigation of the roughness-induced transition: global stability analyses and direct numerical simulations. J. Fluid Mech. 760, 175–211 (2014)
Luchini, P., Bottaro, A.: Adjoint equations in stability analysis. Ann. Rev. Fluid Mech. 46, 493–517 (2014)
Mack, L.: Boundary layer stability theory. Technical report 900-277, Jet Propulsion Laboratory, Pasadena, 1969
Mack, L.: A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73, 497–520 (1976)
Malik, M.R.: Numerical methods for hypersonic boundary layer stability. J. Comput. Phys. 86, 376–413 (1990)
Malik, M.R., Zang, T.A., Hussaini, M.Y.: A spectral collocation method for the Navier-Stokes equations. J. Comput. Phys. 61, 64–88 (1985)
Manneville, P.: Transition to turbulence in wall-bounded flows: where do we stand? Mech. Eng. Rev. 3(2), 15–00684 (2016)
Monokrousos, A., Åkervik, E., Brandt, L., Henningson, D.S.: Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers. J. Fluid Mech. 650, 181–214 (2010)
Nayar, M., Ortega, U.: Computation of selected eigenvalues of generalized eigenvalues problems. J. Comput. Phys. 108, 8–14 (1993)
Orszag, S.A.: Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50, 659–703 (1970)
Pernice, M., Walker, H.F.: NITSOL: a Newton iterative solver for nonlinear systems. SIAM J. Sci. Comput. 19(1), 302–318 (1998)
Peyret, R.: Spectral Methods for Incompressible Viscous Flow. Springer, New York (2002)
Peyret, R., Taylor, T.: Computational Methods for Fluid Flows. Springer, New York (1983)
Pruett, C.D., Gatski, T.B., Grosch, C.E., Thacker, W.D.: The temporally filtered navier-stokes equations: properties of the residual stress. Phys. Fluids 15(8), 2127–2140 (2003)
Pruett, C.D., Thomas, B.C., Grosch, C.E., Gatski, T.B.: A temporal approximate deconvolution model for large-eddy simulation. Phys.Fluids 18(2), 028104 (2006)
Yang, C., Lehoucq, R.B., Sorensen, D.C.: ARPACK user’s guide: Solution of large scale eigenvalue problems with implicitly restarted arnoldi methods. Technical Note, 1997
Rasthofer, U., Wermelinger, F., Hadijdoukas, P., Koumoutsakos, P.: Large scale simulation of cloud cavitation collapse. Procedia Comput. Sci. 108, 1763–1772 (2017)
Renac, Florent: Improvement of the recursive projection method for linear iterative scheme stabilization based on an approximate eigenvalue problem. J. Comput. Phys. 230(14), 5739–5752 (2011)
Richez, F., Leguille, M., Marquet, O.: Selective frequency damping method for steady RANS solutions of turbulent separated flows around an airfoil at stall. Comput. Fluids 132(Supplement C), 51–61 (2016)
Saad, Y.: Iterative Methods for Sparse Linear Systems, vol. 82. SIAM (2003)
Schmid, P.J., Brandt, L.: Analysis of fluid systems: stability, receptivity, sensitivity. Lecture notes from the FLOW-NORDITA summer school on advanced instability methods for complex flows, Stockholm, Sweden, 2013. Appl. Mech. Rev. 66(2), 024803 (2014)
Shaabani-Ardali, L., Sipp, D., Lesshafft, L.: Time-delayed feedback technique for suppressing instabilities in time-periodic flow. Phys. Rev. Fluids 2(11), 113904 (2017)
Shroff, G.M., Keller, H.B.: Stabilization of unstable procedures: the recursive projection method. SIAM J. Numer. Anal. 30(4), 1099–1120 (1993)
Sipp, D., Lebedev, A.: Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358 (2007)
Sorensen, D.C.: Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357–385 (1992)
Stewarson, K., Stuart, J.T.: A non-linear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529–545 (1971)
Stewart, G.W.: A Krylov-Schur algorithm for large eigenproblems. SIAM J. Matrix Anal. Appl. 23, 601–614 (2001)
Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Texts in Applied Mathematics, vol. 12, 3rd edn. Springer, Berlin (2002)
Theofilis, V.: Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 39, 249–315 (2003)
Theofilis, V.: Global linear instability. Ann. Rev. Fluid Mech. 43, 319352 (2011)
Tuckerman, L.S., Barkley, D.: Bifurcation analysis for timesteppers. In: Numerical Methods for Bifurcation Problems and Large-scale Dynamical Systems, pp. 453–466. Springer, Berlin (2000)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Loiseau, JC., Bucci, M.A., Cherubini, S., Robinet, JC. (2019). Time-Stepping and Krylov Methods for Large-Scale Instability Problems. In: Gelfgat, A. (eds) Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics. Computational Methods in Applied Sciences, vol 50. Springer, Cham. https://doi.org/10.1007/978-3-319-91494-7_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-91494-7_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-91493-0
Online ISBN: 978-3-319-91494-7
eBook Packages: EngineeringEngineering (R0)