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

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.

Supraconservative Finite-Volume Methods for the Euler Equations of Subsonic Compressible Flow

“…In search for more 'useful' secondary invariants, in a finite-element setting Layton et al [63] compared several formulations of the equations, but they did not include the conservation form (1.1) which is our starting point. As a follow-up, Rebholz and colleagues [23,24,29] further extended the quest for finite-element methods with enhanced conservation properties, again motivated by accurate long-time integration [8]. Also, Lehmkuhl et al [65] advocate the use of low-dissipative and conservative finite-element schemes.…”

Variational multiscale modeling with discretely divergence-free subscales: Non-divergence-conforming discretizations

“…We leave to future work the study of whether similar empirical results can be obtained using finite element or isogeometric discretizations that are merely inf-sup stable, such as those used in the convergence tests of this paper. In extending our discretization of unsteady Navier-Stokes to high Reynolds number flows, it is also worth exploring alternate formulations of advection, such as the "EMAC" form [31,32], which can retain energetic stability with superior accuracy to the skew form employed in this work. However, a careful comparison of advective forms is beyond the scope of the present study.…”

Analysis of a momentum conservative mixed‐FEM for the stationary Navier–Stokes problem

“…which in the distributional sense means that Δu − ∇p = (∇u)u − f in Ω. Hence, since u belongs to L 4 (Ω), it follows that (∇u)u is in L 4∕3 (Ω), which implies that…”

“…Actually, in most of the software designed to solve partial differential equations, such as Freefem++ and Fenics , the classical families of finite elements for the Stokes problem are already available (see Reference [1] for a detailed study of these classical families). However, it is well‐known that, in general for flow problems, conforming ‐discretizations do not conserve momentum, as it is the case of conforming velocity–pressure discretizations of NS which may cause dissipation of energy and produce a lower bound on the error when approximating the unsteady case (see Reference [2, section 3]), unless the convective term is modified properly, as it is done, for instance, in References [3, 4]. In order to circumvent this lack of momentum conservativity, many researchers have turned to other type of discretizations, such as Finite Volumes and Discontinuous Galerkin methods, among others (see for instance References [5–9], and the references therein).…”