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

Riccardo Milani, Jérôme Bonelle+1 more|arXiv (Cornell University)|May 3, 2021

Advanced Numerical Methods in Computational Mathematics74 references4 citations

TL;DR

This paper proposes a face-based, lowest-order hybrid scheme on polytopal meshes for solving the unsteady incompressible Navier-Stokes equations using artificial compressibility (AC) time integration. The method decouples velocity and pressure updates at each time step, enabling explicit treatment of convection and reducing computational cost by ~50% compared to the monolithic approach, while maintaining second-order accuracy and optimal convergence rates in both velocity and pressure errors on 3D polytopal meshes.

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.

Motivation & Objective

• To develop a robust, low-order face-based scheme for the incompressible Navier-Stokes equations on general polytopal meshes.
• To assess the accuracy and efficiency of artificial compressibility (AC) time integration against the classical monolithic approach.
• To investigate the performance of first- and second-order time schemes with both implicit and explicit treatment of the nonlinear convection term.
• To evaluate the stability and convergence behavior under explicit time stepping, particularly regarding CFL restrictions on Cartesian and polytopal meshes.
• To demonstrate the scalability and computational efficiency of the AC method on large 3D polytopal meshes.

Proposed method

• Uses a lowest-order face-based scheme where discrete velocities are defined on mesh faces and cells, and discrete pressures on mesh cells, enabling support for polytopal meshes.
• Employs artificial compressibility to uncouple velocity and pressure updates: velocity is updated first via a modified Stokes-like problem with a grad-div stabilization term (νη∇∇·u), then pressure is corrected using the divergence of the velocity.
• Applies both first-order (backward Euler) and second-order (BDF2) time discretizations, with explicit or implicit treatment of the convection term.
• Implements the method within the framework of hybrid mixed mimetic and gradient schemes, ensuring consistency and stability on general polytopal meshes.
• Uses iterative linear solvers (GKB and CG) with adaptive tolerance control to balance accuracy and computational cost.
• Employs a bootstrap procedure for second-order AC schemes to reconstruct intermediate states, ensuring temporal accuracy.

Experimental results

Research questions

• RQ1 How does the artificial compressibility method compare in accuracy and efficiency to the monolithic approach for time integration of the incompressible Navier-Stokes equations on polytopal meshes?
• RQ2 What is the impact of explicit versus implicit treatment of the convection term on stability and computational cost, particularly in terms of CFL restrictions?
• RQ3 What are the convergence rates of the velocity and pressure errors for first- and second-order time schemes using the AC method?
• RQ4 How do the performance characteristics of the AC method vary between Cartesian and polytopal meshes under explicit time stepping?
• RQ5 Can second-order AC schemes achieve significantly better accuracy than first-order schemes with only a moderate increase in computational cost?

Key findings

• The AC method achieves computational times approximately 50% lower than the monolithic approach while maintaining comparable accuracy levels on large 3D polytopal meshes.
• Second-order AC schemes reduce velocity errors by a factor of 3.6 and pressure errors by a factor of 2.3 compared to first-order schemes, with computation times less than twice as high.
• The AC method maintains optimal convergence rates in both L2 and H1 norms for velocity and pressure across all tested schemes and mesh types.
• Explicit time stepping exhibits a linear dependence of the CFL restriction on the reciprocal of the Reynolds number, with only slightly tighter constraints on polygonal meshes compared to Cartesian ones.
• The choice of artificial compressibility parameter η = 10 yields better computational efficiency than η = 50, particularly in the first-order AC scheme.
• The bootstrap procedure in second-order AC schemes requires two linear solves per time step, but the overall performance remains favorable compared to monolithic BDF2, which requires only one solve but is significantly more expensive overall.