Computing eigenvalues occuring in continuation methods with the Jacobi-Davidson QZ method

Continuation methods are a well-known technique for computing several stationary solutions of problems involving one or more physical parameters. In order to determine whether a stationary solution is stable, and to detect the bifurcation points of the problem, one has to compute the rightmost eigenvalues of a related, generalized eigenvalue problem. The recently developed Jacobi-Davidson QZ method can be very eective for computing several eigenvalues of a given generalized eigenvalue problem. In this paper we will explain how the Jacobi-Davidson QZ method can be used to compute the eigenvalues needed in the application of continuation methods. As an illustration, the two-dimensional Rayleigh-Benard problem has been studied, with the Rayleigh number as a physical parameter. We investigated the stability of stationary solutions, and several bifurcation points have been detected. The Jacobi-Davidson QZ method turns out to be very ecient for this problem

Stability Analysis of Galerkin/Runge-Kutta Navier-Stokes Discretisations on Unstructured Grids

This paper presents a timestep stability analysis for a class of discretisations applied to the linearised form of the Navier-Stokes equations on a 3D domain with periodic boundary conditions. Using a suitable definition of the `perturbation energy' it is shown that the energy is monotonically decreasing for both the original p.d.e. and the semi-discrete system of o.d.e.'s arising from a Galerkin discretisation on a tetrahedral grid. Using recent theoretical results concerning algebraic and generalised stability, sufficient stability limits are obtained for both global and local timesteps for fully discrete algorithms using Runge-Kutta time integration

Pseudospectra for matrix pencils and stability of equilibria

The concept of "{pseudospectra for matrices, introduced by Trefethen and his co-workers, has been studied extensively since 1990. In this paper, "{ pseudospectra for matrix pencils, which are relevant in connection with generalized eigenvalue problems, are considered. Some properties as well as the practical computation of "{pseudospectra for matrix pencils will be discussed. As an application, we demonstrate how this concept can be used for investigating the asymptotic stability of stationary solutions to time-dependent ordinary or partial dierential equations; two cases, based on Burgers' equation, will be shown

Large deviations in linear control systems with nonzero initial conditions

Research in the transient response in linear systems with nonzero initial conditions was initiated by A.A. Feldbaum in his pioneering work [1] as early as in 1948. However later, studies in this direction have faded down, and since then, the notion of transient process basically means the response of the system with zero initial conditions to the unit step input. A breakthrough in this direction is associated with the paper [2] by R.N. Izmailov, where large deviations of the trajectories from the origin were shown to be unavoidable if the poles of the closed-loop system are shifted far to the left in the complex plane. In this paper we continue the analysis of this phenomenon for systems with nonzero initial conditions. Namely, we propose a more accurate estimate of the magnitude of the peak and show that the effect of large deviations may be observed for different root locations. We also present an upper bound on deviations by using the linear matrix inequality (LMI) technique. This same approach is then applied to the design of a stabilizing linear feedback aimed at diminishing deviations in the closed-loop system. Related problems are also discussed, e.g., such as analysis of the transient response of systems with zero initial conditions and exogenous disturbances in the form of either unit step function or harmonic signal.This work was supported in part by the Mega-Grant of the Russian Federation, project no. 14.Z50.31.0031, the Russian Foundation for Basic Research, projects nos. 14-07-00067-a and 14-08-01230-a, and Portugal grants FCT, COMPETE, QREN, FEDER, project VARIANT (PTDC/MAT/111809/2009)