Implicit-explicit Schemes for Incompressible Flow Problems with Variable Viscosity

SIAM Journal on Scientific Computing, vol. 46, pp. A2660–A2682

Abstract

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.

Authors 3

1. Gabriel R. Barrenechea

   University of Strathclyde

   Affiliation as printed

   Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH UK

2. Ernesto F. Castillo

   Universidad de Santiago de Chile

   Affiliation as printed

   Department of Mechanical Engineering, University of Santiago de Chile, Santiago 9170022 Chile

3. Douglas R. Q. Pacheco corresponding Aachen Corresponding author. Chair for Computational Analysis of Technical Systems Chair of Methods for Model‐based Development in Computational Engineering

   RWTH Aachen University · Norwegian University of Science and Technology

   Affiliation as printed

   Corresponding author. Chair for Computational Analysis of Technical Systems and Chair of Methods for Model-based Development in Computational Engineering, RWTH Aachen University, Aachen 52062 Germany

   Department of Mathematical Sciences, NTNU, Trondheim, Norway