On several occasions when we encountered problems that could not be solved readily by analytical methods, we have referred to results from numerical models. In Chapters 5 and 10 we found, in fact, that analytical solutions to problems of glacier flow could be obtained only when the problems were quite simple. The two numerical methods that are most commonly used in modeling are the finite-difference and finite-element methods.

The analytical methods of calculus are based on taking the limit as intervals over which functions are evaluated are allowed to shrink toward zero. In finite-difference and finite-element models, in contrast, we let these intervals remain finite and assume that the functions describing the variation of parameters across them can be replaced by constants, by linear functions, or by low-order polynomials. The resulting equations turn out to be much simpler than the original differential equations, but because the domain of interest is now broken into many small intervals, one must do a large number of repetitive calculations to obtain a solution for the entire domain. Computers are thus used for all but the simplest numerical calculations. Moreover, the numerical solutions are not necessarily as accurate as analytical ones.

In this chapter, we first describe elementary numerical integration. This leads into some straight forward finite-difference calculations that can be carried out with the use of a spread sheet, a short computer program, or available mathematical software.