Basic Finite-Difference Methods

Abstract

As discussed in the preceding chapter, there are two conceptually different ways to represent continuous functions on digital computers: as a finite set of gridpoint values or as a finite set of series-expansion functions. The grid-point approach is used in conjunction with finite-difference methods, which were widely implemented on digital computers somewhat earlier than the series-expansion techniques. In addition, the theory for these methods is somewhat simpler than that for series-expansion methods. We will parallel this historical development by studying finite-difference methods in this chapter and deferring the treatment of series expansion methods to Chapter 4. Moreover, it is useful to understand finitedifference methods before investigating series-expansion techniques because even when series expansions are used to represent the spatial dependence of some atmospheric quantity, the time dependence is almost always discretized and treated with finite differences.