numerical solution of differential equations
is a subject of fundamental importance.
Mathematicians working in a wide variety of areas, including
continuum mechanics, fluid mechanics,
stochastic differential equations, bifurcation theory,
dynamical systems, and mathematical biology,
all increasingly need expertise in the computational treatment of DEs.
The proposed course provides an overview of three
important topics in the computational solution of DEs,
covering both theoretical aspects and practical computation.
MATLAB is chosen as the programming language in view of its ease of use and its
widespread adoption in research and education.
The course is aimed at first and second year mathematics Ph.D. students
working in any area that requires computational solution of
differential equations;
it assumes a familiarity with numerical analysis
but not a strong background in the subject.
Experience in programming is assumed, but it is not necessary for the students
to be familiar with MATLAB.
Each lecturer will provide the students with MATLAB codes for experimentation
and modification in the laboratory sessions.
Programme
The course will open with a formal welcome on Sunday evening
in Hulme Hall, followed
by an opening lecture on computational differential equations
that will set the scene for the course,
given by
Professor Peter Jimack
of the University of Leeds.
The remaning course lectures and laboratories will be held in the
Ferranti Building on the University of Manchester campus
(building 20 on the
campus map).
Monday morning will consist of an introduction to MATLAB,
given by the course organizer,
with a combined lecture/laboratory session.
Courses 1-3 will start on Monday afternoon,
with one hour introductory lectures.
The remaining 6 half-day sessions contain
a further 6 contact hours per course, split flexibly
between lectures and laboratory sessions.
The laboratory sessions will involve the students working through
example sheets with the aid of the lecturer and two demonstrators.
Lunches will be on campus.
The students will stay in Hulme Hall (20 minutes walk from campus),
where other meals will be taken.
Course Overview
Course 0: A Brief Introduction to MATLAB
(Prof. Nick Higham, University of Manchester)
This mini-course aims to provide the basic grounding in MATLAB that is needed
for Courses 1-3 and also introduces the students to the computing environment
that they will be using.
It treats essentials of MATLAB, including
vectors and matrices,
graphics, control structures and logical tests,
and M-files.
Reference:
Course 1: Spectral Methods
(Prof. Nick Trefethen, University of Oxford)
Spectral methods are numerical methods for ODEs and PDEs based on
global rather than local approximations, e.g., Chebyshev or Legendre polynomial
or trigonometric interpolants as opposed to piecewise linears or
quadratics. For smooth problems posed in simple geometries, a
20-line MATLAB spectral method code can often get the answer to
ten digits of accuracy in a few seconds. The lecturer is well known
for using MATLAB codes to implement spectral methods through his
book
Stochastic differential equations arise in mathematical
models of physical systems which possess inherent noise and uncertainty.
Such models have been used with great success in a variety of application
areas, including biology, epidemiology, mechanics, economics and finance.
This course will give a gentle introduction to SDEs and their numerical
simulation, without assuming a background in probability theory.
The treatment will be at the level of the widely-used article
with extra material on applications, error control and computation
of mean exit times.
Course 3: Finite Element Methods
(Prof. David Silvester, University of Manchester)
Finite element approximation methods are widely used in scientific and
engineering computations. The finite element method is also attractive
from a mathematical perspective because of its rigorous underpinning.
The basic aim here is to develop the theory of a priori and a posteriori
error estimation in the context of elliptic PDEs.
The problems discussed will range in complexity from Poisson's equation
modelling diffusion phenomena to the nonlinear Navier-Stokes equations
modelling steady incompressible fluid flow. The laboratory
sessions will introduce the students to the MATLAB software that the
lecturer has developed in conjunction with his forthcoming OUP book
Accommodation
Accommodation has been arranged in single rooms at
Hulme Hall.
In addition to accommodation, breakfast, lunch and evening meal will
be provided for course residents.
For planning travel to Hulme Hall, see the
location map,
a
larger scale map,
and the
Multimap.
Registration
The registration fee is 100 pounds which, for UK-based research
students, includes the cost of course accommodation and meals.
Participants must pay their own travel costs.
EPSRC-supported students can expect that their registration fees and
travel costs will be met by their
departments from the EPSRC Research Training and Support Grant that is
paid to universities with each studentship award.
The number of participants will be limited and those
interested are encouraged to make an early application.
An online application form is available from the
London Mathematical Society.
The closing date for applications is Friday 15 July 2005.
Further Information
Further information is available from:
Professor Nicholas J. Higham
School of Mathematics
University of Manchester
Sackville St.
Manchester M60 1QD
Tel. +44 (0)161 275 5822
Fax. +44 (0)161 306
3669
higham@maths.man.ac.uk.
Page last modified: August 12, 2005
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