
MINISYMPOSIA 
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, longtime 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 highIndex DAEs
Ned Nedialkov and John Pryce are the authors of DAETS, a C++ code
for solving highindex, any order differentialalgebraic 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 leastsquares 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 DulmageMendelsohn
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 threedimensional,
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
IBM

Title
TBA



