Unconditionally stable, linearised IMEX schemes for incompressible flows with variable density

TL;DR

This paper introduces unconditionally stable, linearised IMEX schemes for incompressible flows with variable density, enabling efficient, decoupled, and stable time integration suitable for complex fluid simulations.

Contribution

It develops and analyzes three new IMEX schemes that treat viscous terms implicitly, allowing for linear, decoupled, and unconditionally stable time discretizations for variable density flows.

Findings

Proved unconditional stability of all schemes.

All systems at each step are linear, avoiding nonlinear solves.

Validated through numerical experiments with finite element methods.

Abstract

For the incompressible Navier--Stokes system with variable density and viscosity, we propose and analyse an IMEX framework treating the convective and diffusive terms semi-implicitly. This extends to variable density and second order in time some methods previously analysed for variable viscosity and constant density. We present three new schemes, both monolithic and fractional-step. All of them share the methodological novelty that the viscous term is treated in an implicit-explicit (IMEX) fashion, which allows decoupling the velocity components. Unconditional temporal stability is proved for all three variants. Furthermore, the system to solve at each time step is linear, thus avoiding the costly solution of nonlinear problems even if the viscosity follows a non-Newtonian rheological law. Our presentation is restricted to the semi-discrete case, only considering the time discretisation. In this way, the results herein can be applied to any spatial discretisation. We validate our theory through numerical experiments considering finite element methods in space. The tests range from simple manufactured solutions to complex two-phase viscoplastic flows.