


Numerical Algorithms for Differential Equations and Dynamical Systems / Algorithmes numériques pour les équations différentielles et les systèmes dynamiques (Org: Tony Humphries, McGill University)
 DAVID COTTRELL, McGill University
A backward analysis of simple collisions

Molecular dynamics simulations often involve the numerical integration
of pairwise particle interactions via a constant stepsize method.
Of primary concern in these simulations is the introduction of errors
in velocity statistics. We consider the simple example of symplectic
Euler applied to twoparticle collisions in 1d governed by linear
restoring force and use backward error analysis to predict these
errors. For nearly all choices of system and method parameters, the
postcollision energy is not conserved and depends upon the initial
conditions of the particles and the stepsize of the method. The
analysis of individual collisions is extended to predict energy growth
in systems of particles in 1d. Generalization of the analysis to
predict evolution of statistics in systems with nonlinear interaction
forces in higher dimensions will be discussed briefly.
 EUSEBIUS DOEDEL, Concordia University, 1455 boulevard de Maisonneuve O.,
Montreal, QC H3G 1M8
Bifurcation of periodic orbits in the Circular Restricted
3Body Problem

The Circular Restricted 3Body Problem describes the motion of a body
of negligible mass (a "satellite") in the presence of an EarthMoon
like system (the "primaries"). This problem has been wellstudied
in the literature, with many important contributions, in view of its
application to spacemission design. In this talk I will describe
some recent results on the classification of certain types of
solutions, namely "elemental" periodic orbits, for all values of the
massratio of the primaries. Some of these orbits are of practical
interest; for example the Genesis mission, which is now on its way
back to Earth for a September arrival, was in a socalled Halo orbit.
I will also describe some of the methods used in our numerical work.
 MARTIN GANDER, McGill University, 805 Sherbrooke Street West, Montreal, CA
Moving Mesh Methods and Energy Conservation

Moving mesh methods are based on an elegant formulation of partial
differential equations coupled with mesh equations. In 1987, a Nobel
Prize winner in physics, Tsung Dao Lee, proposed a discrete variant of
continuous models from continuum and quantum mechanics, which, instead
of using a classical discretization on a fixed mesh, leaves the
location of the mesh nodes as unknowns and uses additional equations,
like energy conservation, to determine both the grid location and the
solution on the corresponding grid. We show for a model problem and
its discrete form proposed in the original publication by Lee that
imposing energy conservation leads to a method for which the discrete
trajectory can cease to exist at perfectly regular points of the
underlying continuous physical model. The results presented are joint
work with Bob Russel from Simon Fraser University.
 WAYNE HAYES, IPST Building 224, University of Maryland, College Park, MD
20742, USA
A Practical Shadowingbased Timestep Criterion for Galaxy
Simulations

The "Butterfly Effect", more formally known as "sensitive
dependence on initial conditions", is exhibited by many nonlinear
dynamical systems from integrated circuits to galaxies. This
sensitivity causes small numerical errors to become exponentially
magnified, leading some to believe that trajectories of such
simulations are the result of nothing but magnified noise. To justify
the reliability of such simulations, we turn to the study of
"shadowing". A shadow is an exact trajectory that stays close
to a numerical trajectory for a long time, even in the face of
sensitive dependence. From the standpoint of physics, a numerical
trajectory that has a shadow can be viewed as an experimental
observation of that shadow, which means that the dynamics observed in
the simulation are real. This is a very strong statement of
simulation reliability. However, verifying the existence of a shadow
formally takes time O(N^{3}), where N is the number of components in
the system. In this talk I will outline how I demonstrated the
existence of shadows of galaxy simulations in which N=10^{8}, without
having to perform O(N^{3}) work.
 TONY HUMPHRIES, McGill University
Numerics and Dynamics

In a dynamical system defined by an ordinary differential equation it
is often the asymptotic behaviour of general initial conditions for
large time that is of interest. However traditional numerical analysis
typically focuses on the solution of a given initial value problem
over a finite time interval, and moreover usually provides error
bounds which grow exponentially in time, and so are not useful in the
dynamical systems context. Over the last two decades there has been an
explosion of work in the "numerics of dynamics" to provide techniques
for numerically studying dynamical systems and for providing rigorous
meanings for some of the pretty pictures which are beyond the scope of
traditional numerical analysis. In this session we will see some of
these techniques including direct methods for computing special
trajectories (e.g. periodic orbits), backward error analysis, stiffness
and adaptive timestepping. In this first general talk we set the
scene by introducing some of the issues that arise, and the techniques
required to tackle them.
Along the way we will show that the backward Euler method is a very
bad method, and also find one or two other surprises.
 NED NEDIALKOV, McMaster University
Solving DifferentialAlgebraic Equations by Taylor Series

We present a method for solving numerically an initialvalue problem
differential algebraic equation (DAE). The DAE can be of highindex,
fully implicit, and contain derivatives of order higher than one.
We do not reduce a DAE to a firstorder, lowerindex form: we solve it
directly by expanding its solution in Taylor series. To compute
Taylor coefficients, we employ the structural analysis of J. Pryce and
automatic differentiation.
Generally, our approach succeeds for any DAE whose sparsity structure
correctly represents its mathematical structure. We show that a
failure occurs if and only if the "system Jacobian" of the DAE is
structurally singular up to roundoff, a situation recognizable in
practice.
This method has been implemented in a C++ code by N. Nedialkov.
Numerical results on several standard test problems show that it is
both efficient and accurate.
This is a joint work with J. Pryce, Royal Military College of Science,
England.
 RAYMOND SPITERI, Dalhousie University, Halifax, NS
A Comparison of Stiffness Detection Methods for InitialValue ODEs

Stiffness is one of the most enigmatic yet pervasive concepts in the
numerical solution of initialvalue ODEs. This is especially true for
ODEs arising from methodoflines discretizations of PDEs. The goal of
stiffness detection methods is to allow software to automatically
choose a stiff or nonstiff integrator, where appropriate. Making the
proper choice of integrator often has a profound effect on the overall
efficiency of the software. However, the many faces of stiffness have
resulted in many definitions, many disagreements, and many methods for
its detection. In this talk, I will compare three methods for
stiffness detection and show their performance on a representative
sample of test problems.

