[Paper Review] A Convergent Staggered Scheme for the Variable Density Incompressible Navier-Stokes Equations

J.‐C. Latché, Khaled Saleh|arXiv (Cornell University)|Mar 23, 2016

Advanced Numerical Methods in Computational Mathematics42 references20 citations

TL;DR

This paper presents a convergent, implicit, staggered finite element scheme for the time-dependent variable density incompressible Navier-Stokes equations using Rannacher-Turek elements and a finite volume-based convection discretization that preserves discrete kinetic energy balance. The key contribution is a rigorous convergence proof showing that solutions of the discrete scheme converge to a weak solution of the continuous problem as the mesh and time step tend to zero, establishing both convergence and the existence of weak solutions via compactness and topological degree arguments.

ABSTRACT

In this paper, we analyze a scheme for the time-dependent variable density Navier-Stokes equations. The algorithm is implicit in time, and the space approximation is based on a low-order staggered non-conforming finite element, the so-called Rannacher-Turek element. The convection term in the momentum balance equation is discretized by a finite volume technique, in such a way that a solution obeys a discrete kinetic energy balance, and the mass balance is approximated by an upwind finite volume method. We first show that the scheme preserves the stability properties of the continuous problem (L $\infty$-estimate for the density, L $\infty$ (L 2)-and L 2 (H 1)-estimates for the velocity), which yields, by a topological degree technique, the existence of a solution. Then, invoking compactness arguments and passing to the limit in the scheme, we prove that any sequence of solutions (obtained with a sequence of discretizations the space and time step of which tend to zero) converges up to the extraction of a subsequence to a weak solution of the continuous problem.

Motivation & Objective

• To develop a stable, consistent, and convergent numerical scheme for the variable density incompressible Navier-Stokes equations.
• To ensure the discrete scheme preserves the kinetic energy balance, crucial for stability and reliability in simulations.
• To prove the convergence of discrete solutions to a weak solution of the continuous problem as the discretization parameters tend to zero.
• To establish the existence of discrete solutions using topological degree theory and compactness arguments.

Proposed method

• Uses an implicit time discretization with a staggered, non-conforming finite element method based on the Rannacher-Turek element for velocity and pressure.
• Discretizes the convection term via a finite volume method on dual cells to preserve a discrete kinetic energy balance.
• Approximates the mass balance using an upwind finite volume method on primal cells, with a reconstruction of dual-cell densities and fluxes to ensure consistency.
• Employs a topological degree argument to prove the existence of discrete solutions for each mesh and time step.
• Applies compactness techniques to extract a convergent subsequence from discrete solutions.
• Passes to the limit in the discrete equations to show that the limit satisfies the weak formulation of the continuous problem.

Experimental results

Research questions

• RQ1 Does the proposed staggered finite element scheme for variable density incompressible Navier-Stokes equations converge to a weak solution as the space and time discretization parameters tend to zero?
• RQ2 Can the scheme preserve the discrete kinetic energy balance, and does this lead to stability estimates for the discrete solution?
• RQ3 Is the existence of a discrete solution guaranteed for each discretization level, and can this be proven without relying on continuous theory?
• RQ4 How does the reconstruction of dual-cell quantities from primal-cell data ensure consistency and mass conservation in the discrete setting?

Key findings

• The scheme preserves the L∞(L2) and L2(H1) estimates for velocity and L∞(L∞) estimate for density, mirroring the stability of the continuous problem.
• The discrete kinetic energy balance is preserved via a finite volume convection discretization on dual cells, ensuring stability and enhancing numerical reliability.
• A topological degree argument guarantees the existence of a discrete solution for each mesh and time step.
• By compactness and weak convergence arguments, any sequence of discrete solutions converges (up to a subsequence) to a weak solution of the continuous Navier-Stokes equations.
• The convergence result is established under minimal assumptions, and the proof does not rely on continuous existence theory, providing a constructive existence proof for weak solutions.
• The analysis extends to variable viscosity models with appropriate stabilization, and discrete Korn’s inequality ensures control of velocity in H1 norm.