Exact factorization technique for numerical simulations of incompressible Navier–Stokes flows

Abstract

Splitting techniques break an ill-conditioned indefinite system resulting from incompressible Navier–Stokes equations into well-conditioned subsystems, which can be solved reliably and efficiently. Apart from the ambiguity regarding numerical boundary conditions for the pressure (and for intermediate velocities, whenever introduced), splitting techniques usually incur splitting errors which reduce time accuracy. The discrete approach of approximate factorization techniques eliminates the need of numerical boundary conditions and restores time accuracy by an approximate inversion of some matrix in the case of semi-implicit time schemes. For linear implicit, non-linear implicit, and higher-order semi-implicit time schemes, however, approximate factorization techniques are laborious. In this paper, we systematically present a new and straightforward exact factorization technique. The main contributions of this work include: (1) the idea of removing the splitting error or the idea of restoring time accuracy for fully discrete systems, (2) the introduction of the pressure-update type and the pressure-correction type of exact factorization techniques for any time schemes, and (3) an analysis of several established techniques and their relations to the exact factorization technique. The exact factorization technique is implemented with a standard second-order finite volume method and is verified numerically.