Structure-Preserving Discretization of Incompressible Fluids

TL;DR

This paper develops a geometric, structure-preserving discretization method for incompressible fluids that maintains key physical invariants and offers improved long-term numerical stability.

Contribution

It introduces a novel Eulerian discretization approach based on finite-dimensional Lie groups that preserves the geometric structure of fluid dynamics.

Findings

Discrete equations preserve Kelvin's circulation theorem

The method exhibits good long-term energy behavior

Numerical experiments demonstrate structure preservation

Abstract

The geometric nature of Euler fluids has been clearly identified and extensively studied over the years, culminating with Lagrangian and Hamiltonian descriptions of fluid dynamics where the configuration space is defined as the volume-preserving diffeomorphisms, and Kelvin's circulation theorem is viewed as a consequence of Noether's theorem associated with the particle relabeling symmetry of fluid mechanics. However computational approaches to fluid mechanics have been largely derived from a numerical-analytic point of view, and are rarely designed with structure preservation in mind, and often suffer from spurious numerical artifacts such as energy and circulation drift. In contrast, this paper geometrically derives discrete equations of motion for fluid dynamics from first principles in a purely Eulerian form. Our approach approximates the group of volume-preserving diffeomorphisms using a finite dimensional Lie group, and associated discrete Euler equations are derived from a variational principle with non-holonomic constraints. The resulting discrete equations of motion yield a structure-preserving time integrator with good long-term energy behavior and for which an exact discrete Kelvin's circulation theorem holds.