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

Nicolás Espinoza-Contreras, Gabriel Barrenechea, Ernesto Castillo, Douglas Pacheco

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.

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

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.

Paper Structure

This paper contains 22 sections, 5 theorems, 63 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model problem
  4. Consistent reformulation
  5. Useful notation, identities and inequalities
  6. Iteration-free IMEX methods
  7. A first-order coupled scheme
  8. A second-order coupled scheme
  9. A first-order fractional-step scheme
  10. On second-order fractional-step schemes
  11. Temporal stability analysis
  12. First-order coupled scheme
  13. Second-order coupled scheme
  14. Fractional-step scheme
  15. Implementation aspects
  16. ...and 7 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: Convergence study for the BDF1 scheme, confirming (at least) first-order convergence for all unknowns.
  • Figure 2: Convergence study for the BDF2 scheme, confirming second-order convergence for all unknowns.
  • Figure 3: Convergence study for the fractional-step scheme, confirming (at least) first-order convergence for all unknowns.
  • Figure 4: Rayleigh--Taylor instability: snapshots of the density field for $t= 0, 1.4, 2.1, 2.45, 2.8$ and $3.15$.
  • Figure 5: Rayleigh--Taylor instability: height of the rising bubble on the right wall.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Lemma 2.1: Discrete Gronwall inequality
  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Lemma 4.1: Density stability
  • Remark 4.1
  • Theorem 4.2: Stability of the first-order coupled scheme
  • ...and 3 more