On the solution of the Neumann Poisson problem arising from a compact differencing scheme using the full multi-grid method

A methodology for the numerical solution of the Neumann-Poisson problem for pressure that arises during the simulation of the incompressible Navier-Stokes equations on non-staggered grids is presented in this study. A sixth order compact differencing scheme has been used for discretizing the governing equation. A general procedure for implementing the discretized form of the integral constraint is proposed. Furthermore, different strategies for handling the corners in a rectangular domain are proposed and evaluated. Solutions for a model problem with a known analytical solution have been obtained on different grids using the full multi-grid technique with Bi-Conjugate Gradient Stabilized method as the smoother. Systematic order studies have been carried out. These bring out the fact that the overall order of the numerical solution is determined by the order of the discretization used for the boundary condition in the case of the Neumann-Poisson problem.