Differential forms, fluids, and finite models

TL;DR

This paper reformulates the Navier-Stokes equations using differential forms, creating finite-dimensional models that approximate fluid dynamics and establishing properties and convergence results for these models.

Contribution

It introduces a novel differential forms approach to finite fluid models and proves approximation properties, bridging continuous equations and finite models.

Findings

Finite models accurately approximate Navier-Stokes equations.

Properties of finite models are systematically derived.

Approximation results demonstrate convergence of finite models.

Abstract

By rewriting the Navier-Stokes equation in terms of differential forms we give a formulation which is abstracted and reproduced in a finite dimensional setting. We give two examples of these finite models and, in the latter case, prove some approximation results. Some useful properties of these finite models are derived.