On the Convergence and Stability of a Pressure–Velocity Iteration of Chorin Type with Natural Boundary Conditions

Abstract

Because of the implementation of numerical solution algorithms for the nonstationary Navier–Stokes equations of an incompressible fluid on massively parallel computers iterative methods are of special interest.

A red–black pressure–velocity iteration that allows an efficient parallelization based on a domain decomposition [3] will be analyzed in this paper.

We prove the equivalence of the pressure–velocity iteration (PUI) by Chorin/Hirt/Cook [1, 2] with a SOR iteration to solve a Poisson equation for the pressure. We show this on a 2D rectangle with some special outflow boundary conditions and Dirichlet data for the velocity elsewhere. This equivalence allows us to prove the convergence of that iteration scheme. We also discuss the stability of the occurring discrete Laplacian in discrete Sobolev spaces.

Keywords:

• Finite volume discretization
• Incompressible viscous Navier–Stokes equation
• Pressure–velocity-iteration
• Semi-implicit time discretization

AMS Subject Classification:

• 65N12
• 65N06
• 76D05
• 65F10
• 65M06