Collocated finite-volume method for the incompressible Navier–Stokes problem

Abstract A collocated finite-volume method for the incompressible Navier–Stokes problem is introduced. The method applies to general polyhedral grids and demonstrates higher than the first order of convergence. The velocity components and the pressure are approximated by piecewise-linear continuous and piecewise-constant fields, respectively. The method does not require artificial boundary conditions for pressure but requires stabilization term to suppress the error introduced by piecewise-constant pressure for convection-dominated problems. Both the momentum and continuity equations are approximated in a flux-conservative fashion, i.e., the conservation for both quantities is discretely exact. The attractive side of the method is a simple flux-based finite-volume construction of the scheme. Applicability of the method is demonstrated on several numerical tests using general polyhedral grids.