Abstract

In this talk we consider the task of designing volume preserving numerical methods for divergence-free ordinary differential equations. This is a difficult task also due to the existence of no-go theorems in spaces of dimension three or higher: there exist no B-series method that is volume preserving for arbitrary divergence-free vector fields. Although a large class of numerical methods (like Runge-Kutta or Taylor-series methods) are B-series methods, this no-go theorem does not exclude that volume can be preserved by other classes of methods, for instance methods based on splitting of the vector field. The talk will be divided in two parts. In the first part, I will introduce a new class of splitting methods based on a monomial basis, a Fourier basis and tensor product of these. In the second part of the talk, I will discuss some recent advancements based on the theory of differential volume forms.