  In this paper, we present a numerical technique for performing Lie advection
of arbitrary differential forms. Leveraging advances in high-resolution finite
volume methods for scalar hyperbolic conservation laws, we first discretize the
interior product (also called contraction) through integrals over Eulerian
approximations of extrusions. This, along with Cartan's homotopy formula and a
discrete exterior derivative, can then be used to derive a discrete Lie
derivative. The usefulness of this operator is demonstrated through the
numerical advection of scalar fields and 1-forms on regular grids.
