Implicit-explicit Schemes for Incompressible Flow Problems with Variable Viscosity

SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2660-A2682, August 2024.
Abstract. This article investigates different implicit-explicit (IMEX) methods for incompressible flows with variable viscosity. The viscosity field may depend on space and time alone or, for example, on velocity gradients. Unlike most previous works on IMEX schemes, which focus on the convective term, we propose also treating parts of the diffusive term explicitly, which can reduce the coupling between the velocity components. We present different IMEX alternatives for the variable-viscosity Navier–Stokes system, analyzing their theoretical and algorithmic properties. Temporal stability is proven for all the methods presented, including monolithic and fractional-step variants. These results are unconditional except for one of the fractional-step discretizations, whose stability is shown for time-step sizes under an upper bound that depends solely on the problem data. The key finding of this work is a class of IMEX schemes whose steps decouple the velocity components and are fully linearized (even if the viscosity depends nonlinearly on the velocity) without requiring any CFL condition for stability. Moreover, in the presence of Neumann boundaries, some of our formulations lead naturally to conditions involving normal pseudotractions. This generalizes to the variable-viscosity case what happens for the standard Laplacian form with constant viscosity. Our analysis is supported by a series of numerical experiments.