Manchester Applied Mathematics and Numerical Analysis Seminars

Spring 1999

March 3, 1999, 4.00 pm

Lecture Theatre OF/B9 Oddfellows Hall (Material Science)

Differential equations on Grassmanian manifolds

Prof. Tom Bridges, Dept. of Maths & Statistics, University of Surrey

While the flow of an ordinary differential equations (ODEs) preserves linear independence, the numerical solution of stiff systems of ODEs, which arise for example in the solution of BVPs by shooting and the computation of Lyapunov exponents, can lose linear independence. Many methods have been proposed to maintain linear independence numerically, the most well-known of which are discrete Gram-Schmidt orthogonalisation and continuous orthogonalisation.

In this talk we approach this problem from a differential geometric point of view. A new definition of orthogonalization is presented: restriction of the ODE to a Stiefel manifold, and this definition leads to a new formulation of continuous orthogonalization, which differs in a precise and interesting geometric way from existing orthogonalization routines. Present orthogonalization methods based on Davey's algorithm are shown to have a different differential-geometric interpretation: restriction of the ODE to a Grassmanian manifold.

This leads us to introduce the concept of a Grassmanian integrator for ODEs, which preserves linear independence and not necessarily orthogonality. Using properties of Grassmanian manifolds and their tangent spaces, a new Grassmanian integrator can be introduced which generalizes Davey's algorithm. Furthermore it is shown that the compound-matrix method is a dual Grassmanian integrator: it uses Pluecker coordinates for integrating on a Grassmanian manifold, and this characterization suggests a new algorithm for constructing the compound matrices.

The theory is applied to the numerical solution of the Orr-Sommerfeld equation in hydrodynamic stability theory.

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For further info contact either Matthias Heil (, Mark Muldoon ( the seminar secretary (Tel. 0161 275 5800).

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