Discrete Lie Advection of Differential Forms

TL;DR

This paper introduces a numerical method for Lie advection of differential forms using finite volume techniques, enabling accurate advection of scalar fields and 1-forms on grids.

Contribution

It develops a discrete Lie derivative operator based on high-resolution finite volume methods, Cartan's formula, and a novel discretization of the interior product.

Findings

Effective advection of scalar fields demonstrated

Accurate advection of 1-forms shown on regular grids

Method leverages advances in hyperbolic conservation laws

Abstract

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.