2
Basic Finite-Difference Methods
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 ear!ier 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.
The derivative of a function f (x) at the point Xo could be defined in any of the
following three ways:
df (xo) = !im f(xo + ßX) - f(xo) ,
dx
lu -> O
ßx
f(xo) - Itx« - Ax)
df ( ) _ 10
,
-
dx
df
Xo -
1m
lu->O
10
1m
ß x->O
-(xo) =
dx
ßx
f(xo + ßX) - f(xo - ß X)
2ßx
.
(2.1)
(2.2)
(2.3)
If the derivative of fex) is continuous at xo, all three expressions produce the
same unique answer. Suppose, however, that f is an approximation to some difD. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics
© Springer Science+Business Media New York 1999