
THEMATIC PROGRAMS 

January 17, 2019  
Numerical and Computational Challenges in Science and EngineeringInformal Working Group on Dynamics of Numerics

Tuesday  Wednesday  Thursday  Friday  
10.0011.00am 

Martin Berz  Wayne Enright 
Informal Discussion and Research  
11.0011.30am  Coffee  
11.3012.30pm  Tony Humphries 


Informal Discussion and Research  
12.302.00pm  Lunch  
2.003.00pm 

Discussion Session "Numerics of Dynamics: What do we gain ?" 

Informal Discussion and Research  
3.003.30pm  Tea  
3.30+pm 
3:30: John Butcher 
Informal Discussion and Research

Informal Discussion and Research

Informal Discussion and Research

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 2step BDF can be shown to be A(a) stable for a~70^{o}, with a suitable restriction on the stepsize ratios.
An improvement on the zerostability result for the 3step BDF shows that stepsize ratios up to (1+5^{½})/2 are possible.
When methods have stability matrices of rank 1, this property can be used to find bounds on the norm of products.
Most current pd ip methods in optimization use a predictor corrector method for path following; we are studying the origins of this from the ODE point of view.
A generalization of the discrete Fourier transform, (GDFT), is used to analyze a short, chaotic time segment of a trajectory of the HenonHeiles 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.
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 (twodimensional 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.
Joint work with E. Van Vleck