Alastair Spence

The numerical analysis of bifurcation problems with application to fluid mechanics

In this review we discuss bifurcation theory in a Banach space setting using the singularity theory developed by Golubitsky and Schaeffer to classify bifurcation points. The numerical analysis of bifurcation problems is discussed and the convergence theory for several important bifurcations is described for both projection and finite difference methods. These results are used to provide a convergence theory for the mixed finite element method applied to the steady incompressible Navier–Stokes equations. Numerical methods for the calculation of several common bifurcations are described and the performance of these methods is illustrated by application to several problems in fluid mechanics. A detailed description of the Taylor–Couette problem is given, and extensive numerical and experimental results are provided for comparison and discussion.

The viscous Cahn-Hilliard equation. I. Computations

The viscous Cahn-Hilliard equation arises as a singular limit of the phase-field model of phase transitions. It contains both the Cahn-Hilliard and Allen-Cahn equations as particular limits. The equation is in gradient form and possesses a compact global attractor A, comprising heteroclinic orbits between equilibria. Two classes of computation are described. First heteroclinic orbits on the global attractor are computed; by using the viscous Cahn-Hilliard equation to perform a homotopy, these results show that the orbits, and hence the geometry of the attractors, are remarkably insensitive to whether the Allen-Cahn or Cahn-Hilliard equation is studied. Second, initial-value computations are described; these computations emphasize three differing mechanisms by which interfaces in the equation propagate for the case of very small penalization of interfacial energy. Furthermore, convergence to an appropriate free boundary problem is demonstrated numerically.

Eigenvalues of Block Matrices Arising from Problems in Fluid Mechanics

Block matrices with a special structure arise from mixed finite element discretizations of incompressible flow problems. This paper is concerned with an analysis of the eigenvalue problem for such matrices and the derivation of two shifted eigenvalue problems that are more suited to numerical solution by iterative algorithms like simultaneous iteration and Arnoldis method. The application of the shifted eigenvalue problems to the determination of the eigenvalue of smallest real part is discussed and a numerical example arising from a stability analysis of double-diffusive convection is described.

Shift-invert and Cayley transforms for detection of rightmost eigenvalues of nonsymmetric matrices

This manuscript is concerned with the determination of the rightmost eigenvalues of large sparse real nonsymmetric matrices. Specifically, the use of subspace iteration preconditioned by the Cayley transform and/or shift-invert is discussed. The convergence properties of subspace iteration are used to construct a strategy to validate the rightmost eigenvalue, which is computed by an iterative method. The motivation behind this paper is that rational preconditioners are very reliable in general but they can miss rightmost eigenvalues with large imaginary part. Numerical examples are given to illustrate the theory.

Eigenvalues of the discretized Navier-Stokes equation with application to the detection of Hopf bifurcations

This paper is concerned with a numerical approach to the problem of finding the leftmost eigenvalues of large sparse nonsymmetric generalised eigenvalue problems which arise in stability studies of incompressible fluid flow problems. The matrices have a special block structure that is typical of mixed finite element discretizations for such problems. The numerical approach is an extension of the hybrid technique introduced by Saad [22] and utilizes the idea of preconditioning the eigenvalue problem before applying Arnoldis method. Two preconditioners, one a modified Cayley transform, the other a Chebyshev polynomial transform, are compared in numerical experiments on a double diffusive convection problem and the Cayley transform proves superior. The Cayley transform is then used to provide numerical results for the finite Taylor problem.

Implicitly restarted Arnoldi with purification for the shift-invert transformation

The need to determine a few eigenvalues of a large sparse generalised eigenvalue problem AZ = λBx with positive semidefinite B arises in many physical situations, for example, in a stability analysis of the discretised Navier-Stokes equation. A common technique is to apply Arnoldis method to the shift-invert transformation, but this can suffer from numerical instabilities as is illustrated by a numerical example. In this paper, a new method that avoids instabilities is presented which is based on applying the implicitly restarted Arnoldi method with the B semi-inner product and a purification step. The paper contains a rounding error analysis and ends with brief comments on some extensions.

Continuation and bifurcations : numerical techniques and applications

Bifurcation to rotating waves from non-trivial steady-states.- Use of approximate inertial manifolds in bifurcation calculations.- Understanding steady-state bifurcation diagrams for a model reaction-diffusion system.- Bifurcations, chaos and self-organization in reaction-diffusion systems.- Eigenvalue problems with the symmetry of a group and bifurcations.- Steady-state/steady-state mode interaction in nonlinear equations with Z2-symmetry.- Symbolic computation and bifurcation methods.- Bifurcation analysis: a combined numerical and analytical approach.- to the numerical solution of symmetry-breaking bifurcation problems.- A computational method and path following for periodic solutions with symmetry.- Global bifurcations and their numerical computation.- Computation of invariant manifold bifurcations.- A method for homoclinic and heteroclinic continuation in two and three dimensions.- The global attractor under discretisation.- The numerical detection of Hopf bifurcation points.- A Newton-like method for simple bifurcation problems with application to large sparse systems.- Aspects of continuation software.- Interactive system for studies in nonlinear dynamics.- LINLBF: A program for continuation and bifurcation analysis of equilibria up to codimension three.- On the topology of three-dimensional separations, A guide for classification.- The construction of cubature formulae using continuation and bifurcation software.- Determining an Organizing Center for Passive Optical Systems.- Optimization by continuation.- Stability of Marangoni convection in a microgravity environment.- Computing with reaction-diffusion systems: applications in image processing.- Bifurcation of Codimension 2 for a discrete map.- Bifurcation of periodic solutions in PDEs: Numerical techniques and applications (abstract).- Bifurcation and chaos in Chuas circuit family (abstract).- Continuation and collocation for parameter dependent boundary value problems (abstract).- Porous medium combustion (abstract).- Block elimination and the computation of simple turning points (abstract).- Bifurcation into gaps in the essential spectrum (abstract).- Application of numerical continuation in aerospace problems (abstract).- Application of a reduced basis method in structural analysis (abstract).- Some applications of bifurcation theory in engineering (abstract).- Higher order predictors in numerical path following schemes (abstract).

Hunting for homoclinic orbits in reversible systems; a shooting technique

A dynamical system is said to be reversible if there is an involution of phase space that reverses the direction of the flow. Examples are Hamiltonian systems with quadratic potential energy. In such systems, homoclinic orbits that are invariant under the reversible transformation are typically not destroyed as a parameter is varied. A strategy is proposed for the direct numerical approximation to paths of such homoclinic orbits, exploiting the special properties of reversible systems. This strategy incorporates continuation using a simplification of known methods and a shooting approach, based on Newtons method, to compute starting solutions for continuation. For Hamiltonian systems, the shooting uses symplectic numerical integration. Strategies are discussed for obtaining initial guesses for the unknown parameters in Newtons method. An example system, for which there is an infinity of symmetric homoclinic orbits, is used to test the numerical techniques. It is illustrated how the orbits can be systematically located and followed. Excellent agreement is found between theory and numerics.

The Numerical Calculation of Cusps, Bifurcation Points and Isola Formation Points in Two Parameter Problems

In this paper we discuss the numerical computation of solutions of the nonlinear, two parameter, problem (1.1) where x ∈ R n is a State variable, λ and α are parameters, and f is a smooth function. Many physical systems can be described by equations like (1.1), see for example, [l],[3],[l2] and [7],[17], where there are more than two parameters.

Inexact Inverse Subspace Iteration with Preconditioning Applied to Non-Hermitian Eigenvalue Problems

Convergence results are provided for inexact inverse subspace iteration applied to the problem of finding the invariant subspace associated with a small number of eigenvalues of a large sparse matrix. These results are illustrated by the use of block-GMRES as the iterative solver. The costs of the inexact solves are measured by the number of inner iterations needed by the iterative solver at each outer step of the algorithm. It is shown that for a decreasing tolerance the number of inner iterations should not increase as the outer iteration proceeds, but it may increase for preconditioned iterative solves. However, it is also shown that an appropriate small rank change to the preconditioner can produce significant savings in costs and, in particular, can produce a situation where there is no increase in the costs of the iterative solves even though the solve tolerances are reducing. Numerical examples are provided to illustrate the theory.