Eulerian and Semi-Lagrangian Methods for Convection-Diffusion for Differential Forms

TL;DR

This paper develops new Eulerian and semi-Lagrangian discretization methods for convection-diffusion problems involving differential forms, with applications to eddy current equations in moving media.

Contribution

It introduces novel discretization techniques for Lie derivatives on differential forms using Galerkin methods, expanding numerical tools for complex physical problems.

Findings

New Eulerian and semi-Lagrangian schemes for differential forms

Application to eddy current equations in moving media

Implementation details and numerical validation

Abstract

We consider generalized linear transient convection-diffusion problems for differential forms on bounded domains in $\mathbb{R}^{n}$. These involve Lie derivatives with respect to a prescribed smooth vector field. We construct both new Eulerian and semi-Lagrangian approaches to the discretization of the Lie derivatives in the context of a Galerkin approximation based on discrete differential forms. Details of implementation are discussed as well as an application to the discretization of eddy current equations in moving media.