# Navier Stokes Project

## October 1999

### Background

The work is concerned with accurate and robust computational
methods for problems having non-smooth solutions. Many of the
grand challenges in modern science and technology give rise to
mathematical problems in this class. Many such problems arise in,
for example, semiconductor device models, petroleum reservoir
models, chemical kinetics, fluid and gas dynamics, liquid crystal
models and mathematical biology. The class includes, in
particular, all singular perturbation problems, which is an
important subclass having a long mathematical history. For
example, laminar flow with large Reynolds number, which is
governed by the Navier-Stokes equations, has been studied for more
than a century.
Modern numerical methods are not effective for solving such
problems in the sense that, as the singular perturbation parameter
becomes small,

- the errors in the numerical solutions increase and often become
comparable in magnitude to the solution itself,
- the number of iterations required to solve the nonlinear algebraic
systems grows unboundedly,
- the conditioning of the algebraic systems deteriorates.

For the above reasons, new theory and computational methods must
be developed for such problems. For this restricted, but important,
class of problems having non-smooth solutions we have developed
numerical methods without rival. This is especially true when the
requisite quantities involve derivatives of the solution, rather
than just the solution itself. In fluid dynamics such quantities
include, for example, the flux and drag, which require
approximations of first order derivatives, and the position of the
separation point, which requires approximations of second order
derivatives.

The main goal of our work is to construct and implement robust
layer-resolving methods for generating numerical approximations
to the solutions and their derivatives of problems of laminar
flow, which are governed by the Navier-Stokes equations. Our
techniques are based on the theoretical work of G. I. Shishkin of
the Russian Academy of Sciences, Ekaterinburg and its detailed
development and implementation by a team of researchers and their
graduate students in Trinity College Dublin, Dublin City
University, University of Limerick and Kent State University,
Ohio, USA. A completely new feature of these methods is that we
can compute realistic estimates of the error parameters for the
method, which lead immediately to realistic estimates of the
parameter-uniform error in the maximum norm of the numerical
solutions generated by these methods.This allows us to specify, in
advance, computational parameters that guarantee any desired
accuracy independently of the value of the Reynolds number.

We believe that within the next four to five
years we will have succeeded in developing methods in the laminar
regime up
to the point of transition to turbulence. This would be a most useful
achievement because so many practical problems are governed by these
equations.

Participating Research Groups

Recent work

Recent Publications

Publications in Press

Presentations at Conferences

Contact Information

Research and
Postgraduate Studies

Maths Department

Trinity College Dublin

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