The Finite Difference Method - Mechanics of Structures: Variational and Computational Methods - page 496

In a similar fashion, beginning with x = x k + 1 , the second order backward difference formulas

are found to be

1
2 h (

u k = u k − 2 − 4 u k − 1 + 3 u k )

(8.6)

1
h 2 (

u k = u k − 2 − 2 u k − 1 + u k )

Other finite difference formulas are derived in the same fashion. Miller (1979) lists central,
forward, backward, and skewed formulas for up to sixth order for uniformly spaced points.

8.1.2 Use of a Taylor Series to Derive Difference Expressions and to Study Truncation Error
A Taylor 1 series can be used to develop the finite difference approximations for the deriva-
tives of the function u

( x ). Taylor's formula appears as

x 2
2! u 0 +···+

x m
m ! u ( m )

x
1! u 0 +

u ( x ) = u 0 + (8.7)

0

For x 1 , the influence of higher powers of x becomes ever smaller. A Taylor series

applies to functions that are continuously differentiable at least up to the level involved in
the remainder.
Suppose the first derivative at point 0 is to be approximated using u 0 and the u values of
neighboring points. Thus, let u 0

<<

be of the form

1
h ( −

u 0 = a

1 u

− a 0 u 0 − a 1 u 1 ) + R

− − 1

or

hu 0 +

R (8.8)
where R is the discretization (truncation) error which will be neglected in the approximate
formula for u 0 .

a

1 u

+ a 0 u 0 + a 1 u 1 = 0 +

− − 1

The Taylor series will yield an estimation of the error in the approximation.
The coefficients a

1 ,a 0 , and a 1 are to be determined such that the error R is made as small
as possible. In the first expression, the minus signs were arbitrarily introduced. In terms of
the Taylor expansion,

−

h 2
2! u 0 −

h 3
3! u

h
1! u 0 +

u = u 0 − +···

− 1

0

h 2
2! u 0 +

h 3
3! u

h
1! u 0 +

u 1 = u 0 + +···

0

Substitution of these relations into Eq. (8.8) gives

h
1! u 0

hu 0 + a

1 u

+ a 0 u 0 + a 1 u 1 = ( a + a 0 + a 1 ) u 0 + ( − a + 1 + a 1 )

− − 1 − 1 − 1

h 2
2! u 0 + ( −

h 3
3! u

+ ( a − 1 + 0 + a 1 ) a − 1 + 0 + a 1 ) +···

0

1 Brook Taylor (1685-1731) was a Cambridge-educated British mathematician who experienced ill health in his
later years. As a product of a brief period of high mathematical productivity, in 1715 he published the two topics
Methodus incrementorum directa et inversa and Linear Perspective . He began to write on religion and philosophy,
resulting in a topic Contemplatio philosophica that appeared posthumously in 1793. In 1712 he wrote a letter to
J. Machin in which he described the expansion of a function in an infinite series, an idea he had during a conver-
sation in a coffee-house. Although there is some dispute as to who discovered “Taylor's series”first, there is little
question that Taylor made his discovery independently.