A direct method for the solution of poisson's equation with neumann boundary conditions on a staggered grid of arbitrary size

Abstract

A method based on cyclic reduction is described for the solution of the discrete Poisson equation on a rectangular two-dimensional staggered grid with an arbitrary number of grid points in each direction. Neumann boundary conditions are assumed in one direction and any boundary condition may be used in the other direction. The coefficients of the equation can be functions of the latter direction so that, e.g., non-equidistant grid spacings or non-Cartesian coordinates can be used. Poisson's equation with these boundary conditions describes, e.g., the pressure field of an incompressible fluid flow within rigid boundaries. Numerical results are reported for a FORTRAN subroutine using the method. For an M × N grid the operation count is proportional to MN log₂ N, and about MN storage locations are required.