Fourth-order finite difference scheme for Poisson's equation in polar coordinates on the unit disc, A

Abstract

We solve Poisson's Equation on the unit disc in polar coordinates using a fourth-order finite difference method. We use a half-point shift in the r direction, in order to avoid approximating the solution at r = 0. We derive a new fourth-order accurate finite difference method from analysis of the truncation error of the well-known second-order scheme. The resulting linear system is solved very efficiently (with cost almost proportional to the number of unknowns) using a combination of a Matrix Diagonalization Algorithm and Fast Fourier Transforms.