Explicit Volume-Preserving Splitting Methods for Linear and Quadratic Divergence-Free Vector Fields

We present new explicit volume-preserving methods based on splitting for polynomial divergence-free vector fields. The methods can be divided in two classes: methods that distinguish between the diagonal part and the off-diagonal part and methods that do not. For the methods in the first class it is possible to combine different treatments of the diagonal and off-diagonal parts, giving rise to a number of possible combinations. This paper is dedicated to Arieh Iserles on the occasion of his 60th anniversary.