An adaptive preconditioner for steady incompressible flows

TL;DR

This paper introduces an adaptive preconditioner for steady incompressible Navier-Stokes flows, improving numerical continuation by smoothly transitioning between no preconditioning and Stokes preconditioning, tested on complex fluid flow models.

Contribution

It presents a novel adaptive preconditioner that interpolates between no preconditioning and Stokes preconditioning for Navier-Stokes equations, enhancing numerical stability and efficiency.

Findings

Mixed preconditioning is preferred for doubly diffusive convection.

Parameter-dependent preconditioning improves solution robustness.

The preconditioner is effective across different fluid flow models.

Abstract

This paper describes an adaptive preconditioner for numerical continuation of incompressible Navier--Stokes flows. The preconditioner maps the identity (no preconditioner) to the Stokes preconditioner (preconditioning by Laplacian) through a continuous parameter and is built on a first order Euler time-discretization scheme. The preconditioner is tested onto two fluid configurations: three-dimensional doubly diffusive convection and a reduced model of shear flows. In the former case, Stokes preconditioning works but a mixed preconditioner is preferred. In the latter case, the system of equation is split and solved simultaneously using two different preconditioners, one of which is parameter dependent. Due to the nature of these applications, this preconditioner is expected to help a wide range of studies.