Application of numerical Lie group integrators to parabolic PDEs

This paper presents a way of solving a parabolic PDE with Runge-Kutta Methods on manifolds as described in papers \cite{bib-munthe-kaas99HoRKmom}, \cite{bib-ActaNumerica99}. These methods are designed to solve ODEs on manifolds using Lie-group theory. An essential step of Runge-Kutta Methods on manifolds is a Lie algebra action which in this case corresponds to solving a simple parabolic PDE. Both Spectral Methods and Finite Difference Methods were tested to obtain approximations to Lie algebra actions. The goal was to develop methods which are more stable than classical Runge-Kutta Methods applied to a parabolic PDE.