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An asymptotic-preserving and exactly mass-conservative semi-implicit scheme for weakly compressible flows based on compatible finite elements

Enrico Zampa, Michael Dumbser

TL;DR

The paper develops an asymptotic-preserving, semi-implicit finite element method for weakly compressible and incompressible flows using compatible FE spaces, with the momentum in an $H(\mathrm{div})$-conforming space to guarantee exact discrete mass conservation. It employs a split-time approach where the convection is discretized explicitly via discontinuous Galerkin, and pressure/viscous terms are treated implicitly, yielding symmetric positive definite linear systems through hybridization. An a posteriori MOOD limiter stabilizes shocks, and the vorticity-based formulation enables efficient decoupling of vorticity from pressure, ensuring an efficient computation path even in the incompressible limit. In the zero-Mach regime, the method converges to an exact divergence-free discretization of the incompressible Navier–Stokes equations, with numerical experiments validating AP behavior and mass conservation across test problems. Overall, the framework provides a high-order, structure-preserving, all-Mach-number capable scheme with favorable computational properties for weakly compressible and incompressible flows.

Abstract

We present a novel asymptotic-preserving semi-implicit finite element method for weakly compressible and incompressible flows based on compatible finite element spaces. The momentum is sought in an $H(\mathrm{div})$-conforming space, ensuring exact pointwise mass conservation at the discrete level. We use an explicit discontinuous Galerkin-based discretization for the convective terms, while treating the pressure and viscous terms implicitly, so that the CFL condition depends only on the fluid velocity. To handle shocks and damp spurious oscillations in the compressible regime, we incorporate an a posteriori limiter that employs artificial viscosity and is based on a discrete maximum principle. By using hybridization, the final algorithm requires solving only symmetric positive definite linear systems. As the Mach number approaches zero and the density remains constant, the method converges to an $H(\mathrm{div})$-based discretization of the incompressible Navier-Stokes equations in the vorticity-velocity-pressure formulation. Several numerical tests validate the proposed method.

An asymptotic-preserving and exactly mass-conservative semi-implicit scheme for weakly compressible flows based on compatible finite elements

TL;DR

The paper develops an asymptotic-preserving, semi-implicit finite element method for weakly compressible and incompressible flows using compatible FE spaces, with the momentum in an -conforming space to guarantee exact discrete mass conservation. It employs a split-time approach where the convection is discretized explicitly via discontinuous Galerkin, and pressure/viscous terms are treated implicitly, yielding symmetric positive definite linear systems through hybridization. An a posteriori MOOD limiter stabilizes shocks, and the vorticity-based formulation enables efficient decoupling of vorticity from pressure, ensuring an efficient computation path even in the incompressible limit. In the zero-Mach regime, the method converges to an exact divergence-free discretization of the incompressible Navier–Stokes equations, with numerical experiments validating AP behavior and mass conservation across test problems. Overall, the framework provides a high-order, structure-preserving, all-Mach-number capable scheme with favorable computational properties for weakly compressible and incompressible flows.

Abstract

We present a novel asymptotic-preserving semi-implicit finite element method for weakly compressible and incompressible flows based on compatible finite element spaces. The momentum is sought in an -conforming space, ensuring exact pointwise mass conservation at the discrete level. We use an explicit discontinuous Galerkin-based discretization for the convective terms, while treating the pressure and viscous terms implicitly, so that the CFL condition depends only on the fluid velocity. To handle shocks and damp spurious oscillations in the compressible regime, we incorporate an a posteriori limiter that employs artificial viscosity and is based on a discrete maximum principle. By using hybridization, the final algorithm requires solving only symmetric positive definite linear systems. As the Mach number approaches zero and the density remains constant, the method converges to an -based discretization of the incompressible Navier-Stokes equations in the vorticity-velocity-pressure formulation. Several numerical tests validate the proposed method.
Paper Structure (21 sections, 3 theorems, 53 equations, 1 table)

This paper contains 21 sections, 3 theorems, 53 equations, 1 table.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Viscous regularization
  4. Asymptotic limit
  5. Numerical method
  6. Time discretization
  7. Space discretization
  8. Notation
  9. Path-conservative DG scheme for the entropy
  10. Convection of the momentum
  11. Mixed finite element discretization of the momentum-pressure system
  12. Efficient decoupling of the vorticity from the pressure
  13. Linear algebra and hybridization
  14. A posteriori limiting via artificial viscosity
  15. Summary of the algorithm
  16. ...and 6 more sections

Key Result

Theorem 3.1

Assume that $\mathbf{m}_h\cdot \mathbf{n}$ and $\mathbf{n} \times \boldsymbol{\omega}_h$ vanish on the boundary of $\Omega$ and $\epsilon_{\rho} = 0$. If the Newton iteration eq:discrete_Newton converges, then the resulting scheme conserves mass locally and momentum globally, that is: Furthermore, if either $\epsilon_{\rho} = 0$ or $\nabla \rho \cdot \mathbf{n}= 0$ on $\partial\Omega$, mass is co

Theorems & Definitions (14)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1
  • proof
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Lemma 3.1
  • ...and 4 more