Symplectic integration algorithms are algorithms for the solution of ordinary differential equations of conservative systems (typically, frictionless systems are conservative.) In a symplectic algorithm the algorithmic time step map exactly preserves a particular mathematical structure exactly as the actual model preserves it. These algorithms have been used advantageously, and sometimes unknowingly, for modelling in such disparate areas as planetary dynamics, fluid dynamics, molecular dynamics, and the dynamics of classical multibody systems. Advantage can be obtained even in dissipative models by splitting the model into dissipative and conservative parts.

Hopefully, better than currently existing numerical algorithms for models of continuous fields, such as nonlinear wave equations, may be obtained by extending these symplectic algorithms to the realm of partial differential equations. I, J.E. Marsden (Caltech), and S.Shkoller (LANL) have recently suggested what "symplectic" might mean and developed some associated algorithms in the context of PDEs arising from variational principles. In this seminar I will give an elementary account of symplectic integration algorithms and indicate our generalization to variational PDE's. These ideas are directly applicable to the nonlinear wave equation and have been tested on the sine-Gordon equation. I will show that our methods have superior performance to currently existing methods for long-time simulations (e.g. over 5000 soliton collisions) of the sine-Gordon equation.

This seminar is the first of a sequence of talks associated with the proposed "Institute of Scientific Modeling" and is co-sponsored by the Department of Computer Science, University of Saskatchewan.