The Averaged Vector Field Method and its Applications to Hamiltonian PDEs

It has been known for some time that the class of energy preserving $B$-series methods is rich \cite{faou}, but concrete practical schemes have been difficult to find. It was recently realized \cite{quispel08ano} that the Averaged Vector Field (AVF) method is an energy preserving $B$-series method. Although this method has been used predominantly for ODEs, we have recently found that it exhibits an excellent behaviour also for certain PDEs. In this talk, we describe the AVF method in general and discuss many of its properties. We give examples of how the AVF method can be applied to PDEs and we describe the derivation of higher order methods. We shall also consider a generalization of the AVF method leading to schemes which are linearly implicit.