Artificial compressibility methods for the incompressible Navier-Stokes equations using lowest-order face-based schemes on polytopal meshes

TL;DR

This paper explores artificial compressibility methods combined with face-based discretization schemes on polytopal meshes for the Navier-Stokes equations, demonstrating improved efficiency and accuracy in 3D simulations.

Contribution

It introduces an uncoupled artificial compressibility approach with face-based discretization for Navier-Stokes, analyzing stability and accuracy on complex meshes.

Findings

AC methods are efficient for large 3D polytopal meshes.

Second-order schemes improve solution accuracy.

Explicit treatment of convection is CFL-limited.

Abstract

We investigate artificial compressibility (AC) techniques for the time discretization of the incompressible Navier-Stokes equations. The space discretization is based on a lowest-order face-based scheme supporting polytopal meshes, namely discrete velocities are attached to the mesh faces and cells, whereas discrete pressures are attached to the mesh cells. This face-based scheme can be embedded into the framework of hybrid mixed mimetic schemes and gradient schemes, and has close links to the lowest-order version of hybrid high-order methods devised for the steady incompressible Navier-Stokes equations. The AC timestepping uncouples at each time step the velocity update from the pressure update. The performances of this approach are compared against those of the more traditional monolithic approach which maintains the velocity-pressure coupling at each time step. We consider both first-order and second-order time schemes and either an implicit or an explicit treatment of the nonlinear convection term. We investigate numerically the CFL stability restriction resulting from an explicit treatment, both on Cartesian and polytopal meshes. Finally, numerical tests on large 3D polytopal meshes highlight the efficiency of the AC approach and the benefits of using second-order schemes whenever accurate discrete solutions are to be attained.