Discrete exterior calculus (DEC) for the surface Navier-Stokes equation

TL;DR

This paper introduces a numerical method using discrete exterior calculus for solving the surface Navier-Stokes equations, achieving second order convergence and handling complex topologies like tori.

Contribution

It develops a covariant, DEC-based discretization for surface fluid flow, including harmonic vector fields, with detailed analysis and comparison to existing methods.

Findings

Demonstrates second order convergence of the scheme.

Shows the method's ability to handle complex topologies like tori.

Provides computational results comparing vorticity-stream function approaches.

Abstract

We consider a numerical approach for the incompressible surface Navier-Stokes equation. The approach is based on the covariant form and uses discrete exterior calculus (DEC) in space and a semi-implicit discretization in time. The discretization is described in detail and related to finite difference schemes on staggered grids in flat space for which we demonstrate second order convergence. We compare computational results with a vorticity-stream function approach for surfaces with genus 0 and demonstrate the interplay between topology, geometry and flow properties. Our discretization also allows to handle harmonic vector fields, which we demonstrate on a torus.