11-15 July 2011

SciCADE 2011

hosted by the Fields Institute,
held at the University of Toronto

New Talent Award

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Numerical differential equations and their applications, I
organized by
Anita Layton, Duke University
Wayne Hayes (Imperial College London)
Ned Nedialkov (McMaster University)

This minisymposium, to be held in honor of Professor Kenneth Jackson's 60th birthday, will survey novel discretization techniques in numerical differential equations that address the computational challenges posed by high index, low regularity, long-time integration, etc. Topics include mathematical finance, simulations of the solar system, computational fluid dynamics, and medical imaging.

Ned Nedialkov,
McMaster University
Improved structural analysis for solving high-Index DAEs
Ned Nedialkov and John Pryce are the authors of DAETS, a C++ code for solving high-index, any order differential-algebraic equations (DAEs). It uses Pryce's structural analysis (SA) theory, and expands the solution in Taylor series using automatic differentiation. DAETS is very effective when high accuracy is required and at solving problems of high index: we have solved artificial DAEs of index up to 47. The consistent initialization of a DAE, one of the difficult problems in DAE solving, is handled naturally in DAETs by solving a least-squares optimization problem. However, the original SA of Pryce may require more initial values than necessary for computing a solution. We develop a method based on the Dulmage-Mendelsohn decomposition for substantially reducing the number of variables needed for consistent initialization. This talk will outline the theory and algorithms behind DAETs and present our improved SA.
Wayne Hayes,
Imperial College London
The interplay of chaos between the inner + outer planets in our Solar System
This talk will consist primarily of pretty pictures displaying the chaotic structure of Lyapunov times in the phase space of our Solar System over a timescale of 200 million years, along with some physical arguments for why the structure exists. We will contrast the picture of the structure seen with the isolated outer (Giant) planets, the isolatde inner (Terrestrial) planets, and the structure resulting from throwing all the planets into the model. We will briefly touch on numerical issues encountered during the thousands of integrations that were performed to generate the pictures.
Anita Layton, Duke University
A method of regularized Stokeslets for periodic boundary conditions
We present a numerical method for simulating the coupled motion of a viscous fluid and an immersed boundary or surface in a three-dimensional, periodic computational domain. The fluid flow is described by the Stokes equations. The method is based on boundary integrals, which compute the flow driven by a periodic array of point forces using the Ewald's summation. The Stokeslets used in the boundary integrals are regularized via a smoothing function to reduce quadrature errors that arise from the singularity of the Stokeslets. Using the regularized Stokeslets, one may compute pressure and velocity field as functions of forcing, or vice versa. Numerical examples are presented which illustrate the properties of the method and its wide applicability.
Robert Enenkel
Title TBA