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Implicit-explicit schemes for incompressible flow problems with variable viscosity

Gabriel R. Barrenechea, Ernesto Castillo, Douglas R. Q. Pacheco

TL;DR

This paper addresses stable and efficient time integration for incompressible flows with variable viscosity by developing and analyzing IMEX schemes that treat parts of the viscous term explicitly. It demonstrates that a generalised Laplacian (GL) viscous formulation yields unconditional stability for both monolithic and fractional-step schemes, enabling decoupled velocity solves even when viscosity is shear-dependent (e.g., Carreau models). The study reveals that naive stress-divergence splits can be unstable, while GL-based variants provide robust stability under realistic smoothness assumptions on the viscosity; a CFL-type condition arises for some fractional-step GL formulations but may be pessimistic in practice. Numerical experiments on temporal convergence, cavity flow, and pulsatile hemodynamics corroborate the theoretical stability results and show accurate, stable solutions for challenging variable-viscosity problems. These findings offer a pathway to efficient, stable solvers for generalized Newtonian fluids in engineering and biomedical applications.

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.

Implicit-explicit schemes for incompressible flow problems with variable viscosity

TL;DR

This paper addresses stable and efficient time integration for incompressible flows with variable viscosity by developing and analyzing IMEX schemes that treat parts of the viscous term explicitly. It demonstrates that a generalised Laplacian (GL) viscous formulation yields unconditional stability for both monolithic and fractional-step schemes, enabling decoupled velocity solves even when viscosity is shear-dependent (e.g., Carreau models). The study reveals that naive stress-divergence splits can be unstable, while GL-based variants provide robust stability under realistic smoothness assumptions on the viscosity; a CFL-type condition arises for some fractional-step GL formulations but may be pessimistic in practice. Numerical experiments on temporal convergence, cavity flow, and pulsatile hemodynamics corroborate the theoretical stability results and show accurate, stable solutions for challenging variable-viscosity problems. These findings offer a pathway to efficient, stable solvers for generalized Newtonian fluids in engineering and biomedical applications.

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.
Paper Structure (21 sections, 7 theorems, 84 equations, 6 figures, 3 tables)

This paper contains 21 sections, 7 theorems, 84 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model problem
  4. Alternative formulations of the viscous term
  5. Useful notation, identities and inequalities
  6. Monolithic schemes
  7. Implicit stress-divergence formulation
  8. Implicit-explicit stress-divergence formulations
  9. Implicit-explicit generalised Laplacian formulation
  10. Fractional-step schemes
  11. IMEX stress-divergence formulation
  12. IMEX generalised Laplacian formulation
  13. Summary of theoretical results
  14. On the fully discrete problem
  15. Matrix structure of the IMEX methods
  16. ...and 6 more sections

Key Result

Lemma 2.1

Let $N\in\mathbb{N}$, and $\alpha,B,a_{n},b_{n},c_{n}$ be non-negative numbers for $n=1,\ldots,N$. Let us suppose that these numbers satisfy Then, the following inequality holds:

Figures (6)

  • Figure 1: Temporal convergence study for the IMEX schemes. The quadratic velocity convergence of the fractional-step methods is probably an initial superconvergence that would break down with further temporal refinement.
  • Figure 2: Lid-driven cavity flow with power-law fluid: temporal evolution of the kinetic energy, confirming that a stress-divergence scheme with fully explicit viscosity can be unstable.
  • Figure 3: Lid-driven cavity flow with power-law fluid: steady-state viscosity field obtained using the IMEX-GL method.
  • Figure 4: Ccomputational domain representing an idealised ICA aneurysm.
  • Figure 5: Carreau fluid through Idealised ICA aneurysm: normalised pressure difference (inlet minus outlet) over one cardiac cycle, for different time-step sizes $\tau$. All three computations are based on the monolithic IMEX scheme in GL form, which allows enforcing the outflow condition \ref{['pseudotraction']} naturally.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Lemma 2.1: Discrete Gronwall inequality
  • Lemma 2.2: Discrete Gronwall lemma, conditional version
  • Remark 3.1
  • Lemma 3.1: Stability of the implicit stress-divergence formulation
  • Remark 3.2
  • Theorem 3.2: Stability of an IMEX stress-divergence formulation
  • Theorem 3.3: Stability of the IMEX generalised Laplacian formulation
  • Theorem 4.1: Stability of an IMEX fractional-step scheme in SD form
  • Theorem 4.2: Stability of the IMEX fractional-step scheme in GL form
  • Remark 4.1
  • ...and 2 more