Computational and Applied Math Proseminar

Department of Mathematics and Statistics, Arizona State University

Tuesday, August 24, 2004, 12:15 p.m. in GWC Room 110

Bojan Orel

Department of Mathematics, University of Ljubljana, Slovenia

Geometric Integration

Abstract Qualitative properties of numerical solutions of ordinary differential equations (ODEs) attracted a lot of attention recently. It was shown that classical numerical methods for solving ODEs are very poor at preserving invariants. Linear multistep formulas can preserve only linear invariants, while some of the implicit Runge-Kutta methods can preserve also quadratic invariants.

The term Geometric integrator means a method that is designed not just to minimize (in the classical sense) the numerical error but also to render correctly the invariants (i.e. the geometry) of the solution.

Since an invariant of an ODE can be interpreted as a homogeneous space of some Lie group, the natural environment for investigating invariant-preserving numerical methods for ODEs is the framework of Lie groups. If we know how to retain Lie-group invariance under discretization, we can also keep the solution on the homogeneous space.

We will review briefly recent advances in numerical methods that advance solutions of ODEs on Lie groups. In particular we will metion Runge-Kutta Munthe-Kaas methods for solving general nonlinear ODEs and Magnus methods for the solution of linear ODEs.

For further information please contact: mittelmann@asu.edu