HHO methods for the incompressible Navier-Stokes and the incompressible Euler equations

Abstract

We propose two Hybrid High-Order (HHO) methods for the incompressible Navier-Stokes equations and investigate their robustness with respect to the Reynolds number. While both methods rely on a HHO formulation of the viscous term, the pressure-velocity coupling is fundamentally different, up to the point that the two approaches can be considered antithetical. The first method is kinetic energy preserving, meaning that the skew-symmetric discretization of the convective term is guaranteed not to alter the kinetic energy balance. The approximated velocity fields exactly satisfy the divergence free constraint and continuity of the normal component of the velocity is weakly enforced on the mesh skeleton, leading to H-div conformity. The second scheme relies on Godunov fluxes for pressure-velocity coupling: a Harten, Lax and van Leer (HLL) approximated Riemann Solver designed for cell centered formulations is adapted to hybrid face centered formulations. The resulting numerical scheme is robust up to the inviscid limit, meaning that it can be applied for seeking approximate solutions of the incompressible Euler equations. The schemes are numerically validated performing steady and unsteady two dimensional test cases and evaluating the convergence rates on ℎ-refined mesh sequences. In addition to standard benchmark flow problems, specifically conceived test cases are conducted for studying the error behaviour when approaching the inviscid limit.

A hybrid high-order scheme for the stationary, incompressible magnetohydrodynamics equations

“…The use of hybrid pressure spaces automatically achieves continuity of the normal components of the velocity and magnetic fields [52]. Moreover, the recent work of [11] shows that robustness with respect to the model parameters can be achieved using this approach. The usage of hybrid spaces for q, r however substantially increase the total number of degrees of freedom (even after considering static condensation); for the MHD model, which is essentially two coupled Navier-Stokes problems, this can lead to quite an expensive system to solve.…”

“…There have been numerous studies of hybrid high-order discretisations of the Stokes [24,9] and Navier-Stokes [10,25] equations. By considering a hybrid pressure space, an HHO discretisation of the Navier-Stokes problem can be devised which locally preserves the conservation of mass of the fluid, and even exhibit robustness in the incompressible Euler limit [11]. The conference proceeding [15] devises a HHO method for a magnetostatics problem, albeit without fully exploiting the principle of high-order reconstruction of HHO methods for the curl operator.…”