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

TL;DR

This paper introduces two novel Hybrid High-Order methods for incompressible fluid equations, emphasizing energy preservation and robustness across viscous and inviscid regimes, validated through diverse numerical tests.

Contribution

The paper presents two distinct HHO schemes with different pressure-velocity coupling strategies, one energy-preserving and the other robust for inviscid flows, advancing numerical methods for fluid dynamics.

Findings

Both methods achieve optimal convergence rates.

The energy-preserving scheme maintains kinetic energy in simulations.

The HLL-based scheme is effective for 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.