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A low-order hybrid method for the variable-density incompressible Navier-Stokes equations

Mathias Dauphin, Daniele A. Di Pietro, Jérôme Droniou, Alexandros Skouras

TL;DR

This work develops a low-order hybrid high-order method for the variable-density incompressible Navier–Stokes equations on highly general meshes, enabling polygonal/polyhedral elements and non-matching interfaces. The scheme uses an upwind discontinuous Galerkin discretisation for density to enforce a discrete maximum principle and a velocity-energetic discretisation for the unsteady term to obtain stable energy estimates. The authors provide a complete theory: existence, uniqueness, a priori bounds, and convergence of the discrete solutions to a weak solution of the continuous problem, complemented by numerical experiments that verify the predicted convergence behavior. The approach extends hybrid high-order methodologies to density-variant flows and demonstrates robust stability and convergence in settings with complex mesh geometries, making it attractive for adaptive and heterogeneous simulations in incompressible flow with variable density.

Abstract

In this work we introduce and analyse a new low-order method for the variable-density incompressible Navier-Stokes equations. The main novelty of the proposed method lies in the support of general meshes, possibly including polygonal or polyhedral elements as well as non-matching interfaces. We carry out a complete analysis, showing stability, existence and uniqueness of a discrete solution, and convergence of the latter to a suitably defined weak solution of the continuous problem. Numerical tests validate the theoretical results.

A low-order hybrid method for the variable-density incompressible Navier-Stokes equations

TL;DR

This work develops a low-order hybrid high-order method for the variable-density incompressible Navier–Stokes equations on highly general meshes, enabling polygonal/polyhedral elements and non-matching interfaces. The scheme uses an upwind discontinuous Galerkin discretisation for density to enforce a discrete maximum principle and a velocity-energetic discretisation for the unsteady term to obtain stable energy estimates. The authors provide a complete theory: existence, uniqueness, a priori bounds, and convergence of the discrete solutions to a weak solution of the continuous problem, complemented by numerical experiments that verify the predicted convergence behavior. The approach extends hybrid high-order methodologies to density-variant flows and demonstrates robust stability and convergence in settings with complex mesh geometries, making it attractive for adaptive and heterogeneous simulations in incompressible flow with variable density.

Abstract

In this work we introduce and analyse a new low-order method for the variable-density incompressible Navier-Stokes equations. The main novelty of the proposed method lies in the support of general meshes, possibly including polygonal or polyhedral elements as well as non-matching interfaces. We carry out a complete analysis, showing stability, existence and uniqueness of a discrete solution, and convergence of the latter to a suitably defined weak solution of the continuous problem. Numerical tests validate the theoretical results.
Paper Structure (20 sections, 12 theorems, 120 equations, 3 figures)

This paper contains 20 sections, 12 theorems, 120 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The continuous problem
  3. Discrete setting
  4. Mesh and basic results
  5. Discrete spaces
  6. Discrete differential operators
  7. Discrete velocity $L^2$-product and norm
  8. Space semi-discrete problem
  9. Viscous term
  10. Convective term
  11. Density advection term
  12. Space semi-discrete problem and main results
  13. Numerical illustration
  14. A reformulation of discrete momentum balance with a view toward consistency
  15. A priori estimates, existence, and uniqueness
  16. ...and 5 more sections

Key Result

Proposition 1

For all $p \in \lbrack 1, \infty \rparen$ if $d = 2$ and all $p \in [1, 6]$ if $d = 3$, it holds

Figures (3)

  • Figure 1: Illustration of the meshes used at the initial refinement step of the convergence analysis
  • Figure 2: Convergence analysis of density and velocity to solution \ref{['eq:guermond.solution']} for three mesh types (triangular, Cartesian and hexagonal depicted in Figure \ref{['fig:meshes']})
  • Figure 3: Integration domain in \ref{['eq:est.int.delta']}.

Theorems & Definitions (28)

  • Proposition 1: Discrete Sobolev inequalities
  • proof
  • Proposition 2: Properties of $\boldsymbol{G}_h$
  • Proposition 3: Sequential compactness of $a_h$
  • proof
  • Proposition 4: Boundedness of $c_h$
  • proof
  • Lemma 5: Consistency of $d_h$
  • proof
  • Lemma 6: Boundedness of $d_h$
  • ...and 18 more