Navier Stokes Project

October 1999


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,

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.

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Research and Postgraduate Studies
Maths Department
Trinity College Dublin

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