On the Interpolation Problem for the Poisson Equation on Collocated Meshes

Abstract

The appearence of unphysical velocities in highly distorted meshes is a common problem in many simulations. In collocated meshes, this problem arises from the interpolation of the pressure gradient from faces to cells. Using an algebraic form for the classical incompressible Navier-Stokes equations, this problem is adressed. Starting from the work of F. X. Trias et. al. [FX.Trias et al. JCP 258: 246-267, 2014], a new approach for studying the Poisson equation obtained using the Fractional Step Method is found, such as a new interpolator is proposed in order to found a stable solution, which avoid the appearence of these unpleasant velocities. The stability provided by the interpolator is formally proved for cartesian meshes and its rotations, using fully-explicit time discretizations. The construction of the Poisson equation is supported on mimicking the symmetry properties of the differential operators and the Fractional Step Method. Then it is reinterpreted using a recursive application of the Fractional Step Method in order to study the system as an stationary iterative solver. Furthermore, a numerical analysis for unstructured mesh is also provided.