Discrete exterior calculus (DEC) for the surface Navier-Stokes equation
For researchers in computational fluid dynamics on surfaces, this work provides a novel discretization that captures topological and geometric effects, though it is incremental as it extends existing DEC methods to a new equation.
The paper presents a numerical method for the incompressible surface Navier-Stokes equation using discrete exterior calculus, achieving second-order convergence on flat surfaces and handling harmonic vector fields on tori.
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.