Efficient discretizations for the EMAC formulation of the incompressible Navier-Stokes equations

TL;DR

This paper investigates discretization strategies for the EMAC formulation of the incompressible Navier-Stokes equations, demonstrating that Newton linearization effectively preserves conservation laws and yields accurate long-term solutions.

Contribution

It introduces and compares linearization techniques for EMAC discretizations, highlighting the effectiveness of Newton linearization in conserving physical quantities.

Findings

Newton linearization conserves momentum and angular momentum.

Two Newton steps per time step effectively preserve all conservation laws.

Skew-symmetrized linearization is less accurate than Newton linearization.

Abstract

We study discretizations of the incompressible Navier-Stokes equations, written in the newly developed energy-momentum-angular momentum conserving (EMAC) formulation. We consider linearizations of the problem, which at each time step will reduce the computational cost, but can alter the conservation properties. We show that a skew-symmetrized linearization delivers the correct balance of (only) energy and that the Newton linearization conserves momentum and angular momentum, but conserves energy only up to the nonlinear residual. Numerical tests show that linearizing with 2 Newton steps at each time step is very effective at preserving all conservation laws at once, and giving accurate answers on long time intervals. The tests also show that the skew-symmetrized linearization is significantly less accurate. The tests also show that the Newton linearization of EMAC finite element formulation compares favorably to other traditionally used finite element formulation of the incompressible Navier-Stokes equations in primitive variables.