A natural radial scheme for laplace's equation

Abstract

For two dimensional problems with radial symmetry, the standard numerical scheme applied to Laplace's equation uses the polar form of the equation together with a radial grid based upon equal increments of angle and equal increments of radius. However the resulting numerical scheme contains a fundamental difficulty—it includes an r ^-1 component which can lead to computational inaccuracy near the origin. In this paper a numerical scheme is devised with radial points arranged in non-uniform geometric sequence. The resulting finite difference scheme for Laplace's equation is remarkably concise, is well-behaved at the origin, and the corresponding grid is a natural radial analogue of the standard rectangular grid. Analytical comparisons are made with the standard scheme and computational experiments on an example from electromagnetics indicate significant improvement in accuracy.

Keywords:

• Finite differences
• derivative approximations
• partial differential equations
• elliptic equations
• radial grids