Introduction: basic concepts

Finite difference (FD) approximations for derivatives were already in use by Euler (1768). The simplest FD procedure for dealing with the problem dx/dt = f (t, x), x(0) = x0 is obtained by replacing (dx/dt)n-1 with the crude approximation (xn- xn-1)/Δt, Δt = tn-tn-1. This leads to the recurrence relation xn = xn-1 +Δtf (tn-1, xn-1) for n > 0. This procedure is known as the Euler method (see Section 7.2 for more detail). Therefore, we see that for one-dimensional (1-D) problems the FD approach has been deeply ingrained in computational algorithms for quite some time. For two-dimensional (2-D) problems the first computational application of FD methods was most probably carried out by Runge (1908). He studied the numerical solution of Poisson's equation Δu =δ2u/δx2 +δ2u/δy2 = c, where c is a constant. A few years later Richardson (1910) published his work on the application of iterative methods to the solution of continuous equilibrium problems by FDs. The celebrated paper by Courant et al. (1928) is often considered as the birth date of the modern theory of numerical methods for partial differential equations.

The goal of the FD method is to reduce the ordinary or partial differential equations to discrete equations approximating the differential equations and making then suitable for computer implementation.