Stability of a colocated finite volume scheme for the incompressible Navier-Stokes equations

TL;DR

This paper presents a colocated finite volume scheme for 2D incompressible Navier-Stokes equations on triangular meshes, proving its stability through discrete operator properties and an inf-sup condition.

Contribution

It introduces a stable colocated finite volume scheme for Navier-Stokes equations with a projection method and demonstrates its stability via discrete operator analysis.

Findings

Scheme is stable under the inf-sup condition

Discrete operators mimic continuous properties

Applicable to triangular meshes

Abstract

We introduce a finite volume scheme for the two-dimensional incompressible Navier-Stokes equations. We use a triangular mesh. The unknowns for the velocity and pressure are both piecewise constant (colocated scheme). We use a projection (fractional-step) method to deal with the incompressibility constraint. We prove that the differential operators in the Navier-Stokes equations and their discrete counterparts share similar properties. In particular, we state an inf-sup (Babuska-Brezzi) condition. We infer from it the stability of the scheme.