Stability analysis of pressure correction schemes for the Navier-Stokes equations with traction boundary conditions

TL;DR

This paper analyzes the stability of two pressure correction schemes for Navier-Stokes equations with traction boundary conditions, establishing conditions for unconditional and conditional stability through theoretical analysis.

Contribution

It provides the first stability analysis of rotational pressure correction schemes with traction boundary conditions, including an unconditional stability result for a stabilized scheme.

Findings

The rotational scheme is unconditionally stable with proper stabilization.

A conditional stability result is established for the boundary correction scheme.

Stability is proven using the equivalence between gauge Uzawa methods and pressure correction schemes.

Abstract

We present a stability analysis for two different rotational pressure correction schemes with open and traction boundary conditions. First, we provide a stability analysis for a rotational version of the grad-div stabilized scheme of [A. Bonito, J.-L. Guermond, and S. Lee. Modified pressure-correction projection methods: Open boundary and variable time stepping. In Numerical Mathematics and Advanced Applications - ENUMATH 2013, volume 103 of Lecture Notes in Computational Science and Engineering, pages 623-631. Springer, 2015]. This scheme turns out to be unconditionally stable, provided the stabilization parameter is suitably chosen. We also establish a conditional stability result for the boundary correction scheme presented in [E. Bansch. A finite element pressure correction scheme for the Navier-Stokes equations with traction boundary condition. Comput. Methods Appl. Mech. Engrg., 279:198-211, 2014]. These results are shown by employing the equivalence between stabilized gauge Uzawa methods and rotational pressure correction schemes with traction boundary conditions.