Modern numerical methods are not effective for solving such
problems in the sense that, as the singular perturbation parameter
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
Publications in Press
Presentations at Conferences
Trinity College Dublin
Top of Page