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Time-Stepping and Krylov Methods for Large-Scale Instability Problems

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Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 50))

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.

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Notes

  1. 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. 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. 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.

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Authors and Affiliations

  1. Laboratoire DynFluid, Arts et Métiers ParisTech, 151 boulevard de l’hôpital, 75013, Paris, France

    J.-Ch. Loiseau, M. A. Bucci & J.-Ch. Robinet

  2. DMMM, Politecnico di Bari, via Re David 200, 70100, Bari, Italy

    S. Cherubini

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  1. J.-Ch. Loiseau
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    Alexander Gelfgat

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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

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