SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

# Invited Talk

### Which methods have a B-Series expansion?

O. Verdier and H. Munthe-Kaas

Abstract
Runge-Kutta methods are {affine equivariant} (they are compatible with affine variable transformations), and {local} (they depend only on an infinitesimal neighbourhood of every point). The same holds for B-Series in general. We address the following question: \begin{center} are B-Series the only affine equivariant and local methods? \end{center} The answer turns out to be true in one dimension, and {false} in more dimensions. The methods which are affine equivariant and local, however, {almost} have a B-Series expansion. This observation leads to a new family of methods, which contains Runge-Kutta methods. Those methods have advantages over Runge-Kutta methods: for instance, they can potentially be volume preserving, whereas B-Series are known not to be. We will explain what Runge-Kutta methods, B-Series, equivariance and locality mean in detail, so no prerequisite is required to attend the talk.

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