PolyMAC: A Staggered Finite Volume Method on General Meshes for Incompressible Navier-Stokes Equations

We consider the numerical resolution of the incompressible Navier-Stokes equations. We present a new compatible Finite Volume discretisation that generalises the famous Marker-and-Cell (MAC) method to polyhedral meshes that we call PolyMAC. In the first part of the paper, we recall the principles of compatible schemes and detail the key operators of the discretisation. The convergence and robustness of PolyMAC are assessed numerically first on a benchmark from the FVCA conferences. We then consider a problem of industrial complexity that allows us to confirm the robustness of PolyMAC on more complex problems. The second part of the article is dedicated to the efficient numerical resolution of the resulting linear system. Concretely, we use a PISO-like prediction-correction approach and develop efficient preconditioners for linear systems. In particular, we show that the saddle-point system arising from the correction step is very challenging for iterative methods on distorted meshes. In this work, we develop a robust preconditioner based on an algebraic transformation of the system. In particular, this new preconditioner shows impressive convergence on problems of industrial complexity.