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Consistent splitting SAV schemes for finite element approximations of incompressible flows

Douglas R. Q. Pacheco

TL;DR

The paper addresses stable, high-order consistent splitting schemes for incompressible flows discretized with standard $C^0$ finite elements, where traditional projection-based methods struggle with splitting errors and inf-sup requirements. It introduces a scalar auxiliary variable (SAV) coupled with an IMEX time-stepping and a velocity split, achieving unconditional stability for a second-order $BDF2$-based scheme while keeping per-step work to two linear transport-diffusion problems plus a Poisson solve. Theoretical contributions include a rigorous unconditional temporal stability proof and a pressure bound in a strong norm, complemented by a practical four-step algorithm and comprehensive numerical validation (temporal convergence, stability, and vortex-shedding benchmarks). The approach enables accurate, robust incompressible-flow simulations with standard finite elements and open boundaries, removing the need for inf-sup-stable pairs and enabling large time steps with high fidelity.

Abstract

Consistent splitting schemes are among the most accurate pressure segregation methods, incurring no splitting errors or spurious boundary conditions. Nevertheless, their theoretical properties are not yet fully understood, especially when finite elements are used for the spatial discretisation. This work proposes a simple scalar auxiliary variable (SAV) technique that, when combined with standard finite elements in space, guarantees unconditional stability for first- and second-order consistent splitting schemes. The framework is implicit-explicit (IMEX) and only requires solving linear transport equations and a pressure Poisson problem per time step. Furthermore, pressure stability is attained with respect to a stronger norm than in classical projection schemes, which allows eliminating the inf-sup compatibility requirement on the velocity-pressure pairs. The accuracy of the new framework is assessed through numerical examples.

Consistent splitting SAV schemes for finite element approximations of incompressible flows

TL;DR

The paper addresses stable, high-order consistent splitting schemes for incompressible flows discretized with standard finite elements, where traditional projection-based methods struggle with splitting errors and inf-sup requirements. It introduces a scalar auxiliary variable (SAV) coupled with an IMEX time-stepping and a velocity split, achieving unconditional stability for a second-order -based scheme while keeping per-step work to two linear transport-diffusion problems plus a Poisson solve. Theoretical contributions include a rigorous unconditional temporal stability proof and a pressure bound in a strong norm, complemented by a practical four-step algorithm and comprehensive numerical validation (temporal convergence, stability, and vortex-shedding benchmarks). The approach enables accurate, robust incompressible-flow simulations with standard finite elements and open boundaries, removing the need for inf-sup-stable pairs and enabling large time steps with high fidelity.

Abstract

Consistent splitting schemes are among the most accurate pressure segregation methods, incurring no splitting errors or spurious boundary conditions. Nevertheless, their theoretical properties are not yet fully understood, especially when finite elements are used for the spatial discretisation. This work proposes a simple scalar auxiliary variable (SAV) technique that, when combined with standard finite elements in space, guarantees unconditional stability for first- and second-order consistent splitting schemes. The framework is implicit-explicit (IMEX) and only requires solving linear transport equations and a pressure Poisson problem per time step. Furthermore, pressure stability is attained with respect to a stronger norm than in classical projection schemes, which allows eliminating the inf-sup compatibility requirement on the velocity-pressure pairs. The accuracy of the new framework is assessed through numerical examples.

Paper Structure

This paper contains 18 sections, 1 theorem, 36 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and useful identities and inequalities
  4. The incompressible Navier--Stokes system
  5. Equivalent reformulation
  6. Numerical method
  7. Weak PPE
  8. Temporal discretisation
  9. Solution algorithm
  10. Temporal stability analysis
  11. Comparison with some classical methods
  12. Pressure-stabilised Petrov--Galerkin (PSPG)
  13. Incremental pressure correction
  14. Numerical examples
  15. Temporal convergence test
  16. ...and 3 more sections

Key Result

Theorem 4.1

For any time-step size $\tau=T/N$, the fractional-step scheme BDF2momentumWeak--BDF2PPEweak, with $\hbox{\boldmath $f$}=\hbox{\boldmath $0$}$, satisfies

Figures (4)

  • Figure 1: Taylor--Green vortex: temporal convergence considering a fine mesh with either Taylor--Hood (left) or equal-order (right) elements.
  • Figure 2: Lid-driven cavity flow: evolution of the SAV considering different types of elements.
  • Figure 3: Cylinder flow benchmark: velocity magnitude fields for $t=2,4,6,8$.
  • Figure 4: Cylinder flow benchmark: temporal evolution of the drag coefficient $C_D$, the lift coefficient $C_L$, and the auxiliary variable $\psi$.

Theorems & Definitions (3)

  • Remark 3.1
  • Theorem 4.1: Temporal stability of the second-order scheme
  • Remark 4.1