Decoupled and unconditionally stable iteration method for stationary Navier–Stokes equations

Abstract

It is well known the Oseen iteration for the stationary Navier–Stokes equations is unconditionally stable. However, it is a coupled type scheme where the velocity $u$ and pressure $p$ are coupled together at each iteration. By treating pressure $p$ explicitly would lead to a decoupled iteration, but this treatment is unstable. In this article, we construct a decoupled and unconditionally stable iteration method to solve the stationary Navier–Stokes equations by adopting the pressure projection method to the temporal disturbed Navier–Stokes system whose solution approximates the steady state solution over time ($t\to +\infty$). We also rigorously prove its unconditional stability. Numerical simulations demonstrate that our iterative method is more efficient and stable than the extensively used T-S and Oseen iterations, and could solve the fluid flow with high Reynolds number.