Stability of a colocated finite volume scheme for the incompressible Navier-Stokes equations
Provides a theoretical stability guarantee for a practical numerical scheme, addressing a known bottleneck in colocated finite volume methods for incompressible flows.
The paper introduces a colocated finite volume scheme for the 2D incompressible Navier-Stokes equations on triangular meshes and proves its stability via an inf-sup condition.
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.