Page 72 - Standard Handbook Of Petroleum & Natural Gas Engineering

Numerical Methods 61

The second forward difference of f(x) at xi is
A*fi = fi+' - 2fi+, + f,

and the second derivative of f(x) is then given by
f"(x) = (A2fi)/h2 + O(h)

The second backward difference of f at xI is

V'f, = f, - 2f 3-1 + f,-2
and f"(x) may also be defined as
f"(x) = (V2f,)/h2 + O(h)

Approximate expressions for derivatives of any order are given in terms of
forward and backward difference expressions as

fb) = (A"f,)/h" + O(h) = (V"f,)/h" + O(h)

Coefficients of forward difference expressions for derivatives of up to the
fourth order are given in Figure 1-52 and of backward difference expressions
in Figure 1-53.
More accurate difference expressions may be found by expanding the Taylor
series. For example, f'(x) to V(h) is given by forward difference by

f'(x) = (-fi+, + 4fi+, - 3fi)/(2h) + O(h2)
and a similar backward difference representation can also be easily obtained.
These expressions are exact for a parabola. Forward and backward difference
expressions of O(h2) are contained in Figures 1-54 and 1-55.
A central difference expression may be derived by combining the equations for
forward and backward differences.

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Figure 1-52. Forward difference coefficients of o(h).