Implicit-explicit schemes for incompressible flow problems with variable viscosity

TL;DR

This paper investigates various implicit-explicit schemes for incompressible flows with variable viscosity, focusing on explicit treatment of diffusive terms to improve computational efficiency and analyzing their stability properties.

Contribution

It introduces new IMEX schemes for variable-viscosity Navier-Stokes equations, with comprehensive stability analysis and numerical validation.

Findings

All schemes are unconditionally stable except one with a CFL condition.

Explicit treatment of diffusive terms reduces coupling and computational complexity.

Numerical experiments confirm theoretical stability results.

Abstract

In this work we study different Implicit-Explicit (IMEX) schemes for incompressible flow problems with variable viscosity. Unlike most previous work on IMEX schemes, which focuses on the convective part, we here focus on treating parts of the diffusive term explicitly to reduce the coupling between the velocity components. We present different, both monolithic and fractional-step, IMEX alternatives for the variable-viscosity Navier--Stokes system, analysing their theoretical and algorithmic properties. Stability results are proven for all the methods presented, with all these results being unconditional, except for one of the discretisations using a fractional-step scheme, where a CFL condition (in terms of the problem data) is required for showing stability. Our analysis is supported by a series of numerical experiments.