TABLE OF CONTENTS Finite difference methods have been used to find approximate solutions to differential equations governing physical phenomena for over 100 years. In [Richardson, 1911], finite differences (more precisely, a central difference method [Sheppard, 1899]) were used to calculate the stress within a dam of irregular cross-section. In Richardson's paper he notes,
"The object of this paper is to develop methods whereby the differential equations of physics may be applied more freely than hitherto in the approximate form of difference equations concerning irregular bodies . analytical methods are the foundation of the whole subject, and in practice they are most accurate when they will work, but in the integration of partial (differential) equations, with reference to irregular-shaped boundaries, their field of application is very limited."
More recently, [von Neumann and Richtmyer, 1947] describes both explicit and implicit difference systems for the solution of partial differential equations of parabolic type. They also presented a detailed Fourier stability analysis of both systems. In these papers, and in all subsequent work, the finite difference method uses the Taylor series representation of the derivative as its foundation. In developing the difference method here, much of the information presented draws from books by [Smith, 1965], [Collatz, 1966], and [Tannehill et al., 1997], which are representative of a large body of finite difference literature.
Suppose u = u (x, y). Then the derivative of u with respect to x at the point (x, y) = (x 0, y 0
