Factorization of the Fourier transform of the pressure-Poisson equation using finite differences in colocated grids

The zero-divergence constraint on the velocity field in the numerical simulation of incompressible flows can be reduced, in certain cases, to a set of one-dimensional linear difference equations for the pressure. These equations involve the secondorder derivative dxdxp expressed in terms of twice the first-order derivative. When implicit finite-difference schemes are used, those equations lead to full linear systems, which are computationally prohibitive. Hence, it is a common practice to substitute dxdxp by a different discretization dxxp. However, it is well known that this step results in a non-zero divergence in the velocity field. This paper presents a factorization of the original equation that allows to satisfy the discrete solenoidal constraint exactly while maintaining a linear relation between the number of operations and the grid size. As an example, the method is particularized to compact schemes often found in the literature.