April 16, 2014

Numerical and Computational Challenges in Science and Engineering

Informal Working Group on Dynamics of Numerics
August 7 - 10, 2001

Tuesday Wednesday Thursday Friday
10:00: Welcome: Tony Humphries
10:05: John Butcher
Martin Berz Wayne Enright
Informal Discussion and Research
11.00-11.30am Coffee
11.30-12.30pm Tony Humphries
11:30: Kyoko Makino
12:10: Chris Elmer
11:30: Leo Tzou
12:00: Martin Berz
Part II
Informal Discussion and Research
12.30-2.00pm Lunch
2:00: Allison Heard
2:45: Tom Fairgrieve
Discussion Session "Numerics of Dynamics: What do we gain ?"
2:00: John Grindlay
2:40: Greg Lewis
Informal Discussion and Research
3.00-3.30pm Tea

3:30: John Butcher
Part II
4:00: Informal Discussion and Research

Informal Discussion and Research
Informal Discussion and Research
Informal Discussion and Research

Tony Humphries
University of Sussex

Dynamics of Runge-Kutta Methods

John Butcher,
University of Auckland

Stable and efficient methods for stiff differential equations

Allison Heard,
University of Auckland

Stability of Variable Stepsize Methods

Joint work with JC Butcher.

Most of the work on the stability of numerical methods for ordinary differential equations assumes constant step size. In that case stability involves powers of the stability matrix, but that is not sufficient for the case of variable step size. I will review some approaches to bounding products of matrices from a given class.

The 2-step BDF can be shown to be A(a) stable for a~70o, with a suitable restriction on the stepsize ratios.

An improvement on the zero-stability result for the 3-step BDF shows that stepsize ratios up to (1+5½)/2 are possible.

When methods have stability matrices of rank 1, this property can be used to find bounds on the norm of products.

Leo Tzou and Henry Wolkowicz,
University of Waterloo

A view of predictor corrector methods for primal-dual interior-point methods in optimization using ODEs

Most current p-d i-p methods in optimization use a predictor corrector method for path following; we are studying the origins of this from the ODE point of view.

John Grindlay,
University of Waterloo

A Generalization of the Discrete Fourier Transform applied to the Henon-Heiles Model

A generalization of the discrete Fourier transform, (GDFT), is used to analyze a short, chaotic time segment of a trajectory of the Henon-Heiles model. The dynamic behaviour in this time segment can be well represented by four sinusoids, with characteristic frequencies - two subharmonics 2/5, 3/5 a fundamental 1 and a harmonic 2.

Martin Berz,
Michigan State University

An overview of High-Order Verified Methods and Applications to Symplectic Integration

Kyoko Makino
University of Illinois

Optimal Control of the Wrapping Effect in Taylor Model based Verified Integration

Greg Lewis
University of British Columbia

Dynamics in a differentially heated rotating fluid

I will describe a bifurcation analysis of a differentially heated rotating fluid annulus. Many experiments have been performed on this system, and a rich variety of dynamical behaviour has been observed that has not been theoretically explained. In the talk I would describe the analysis of the double Hopf bifurcations that occur along the transition from steady flow to wave motion. To deduce the dynamics close to the bifurcation points, a center manifold reduction is performed and the coefficients of the normal form equations are calculated. Of particular interest is that the analysis cannot be completed analytically. Discretization of the relevant (two-dimensional partial differential) equations leads to numerical approximations of the normal form coefficients. The most numerically intensive step is the calculation of the eigenvalues and eigenfunctions, that are (after discretization) approximated from a generalized eigenvalue problem with large sparse matrices.

Ideally, I would like to extend the analysis to bifurcations from the waves. It is expected that these bifurcations would be observed as bifurcations from periodic orbits to invariant tori. However, I think that, at this time, this analysis for the fluid annulus would be computationally prohibitive. So for future work I'm planning to investigate a simplified model of a differentially heated rotating fluid.

Tom Fairgrieve
University of Toronto

Normalizing Periodic Solutions

Chris Elmer
New Jersey Institute of Technology

Some effects of discretization on bistable reaction-diffusion equations

Joint work with E. Van Vleck

Wayne Enright
University of Toronto

Useful Tools for Verifying Solutions of ODEs