˜ B ¤£ ˜

¡

˜

order of where is the total number of discretization points. This section gives

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uu u ™

n

¢

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§¥ © ©

an overview of ¬nite difference discretization techniques.

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The simplest way to approximate the ¬rst derivative of a function at the point is via the

V

formula

—

9 BV BV 3

¦ ¦

Y¥

t

C

V B ¨

2&

¦

C

9 ¦

C

¦

When is differentiable at , then the limit of the above ratio when tends to zero is the

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¢

–

¦ ¦

derivative of at . For a function that is in the neighborhood of , we have by Taylor™s

V

formula

69 6

¢9 ¢

9 V9

¢ Y¥

t

— 9 — C B C —

V 6V ¦ 9 4 B V

— —

BV ¨

¢&

¦

¦

£

V ¢¦9

9

¦ ¦

C

— x B

4 ©

¦¦

for some in the interval . Therefore, the above approximation (2.8) satis¬es

C B V69

— B 99 BV 3 6B¡

"

¦

¦ "

¦

Y

t

C —

V V ¨ £ ¦ &¥

3 C 6¦9

C C

¦

3

The formula (2.9) can be rewritten with replaced by to obtain

66

¢9 ¢

— 99 V6 9 9 B V

¢ Y

t

C C

V V

—

¦B V BV ¨ ¦ ¦ &¥

3¦ 3 3

¦ &

£ ¦

¦9 9 ¢¦9

¦

C

B 3

¦

¦

in which belongs to the interval . Adding (2.9) and (2.11), dividing through

6 & C

by , and using the mean value theorem for the fourth order derivatives results in the

following approximation of the second derivative

B V69 ¢9 6

l C — —

BV B BV B V

3 3

C ¢ 9 e C

¦

¦ ¦ ¦

Y

t

6V ¨ & ¦ &¥

3

C 6¦9 C

¦

4

where . The above formula is called a centered difference approximation of

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the second derivative since the point at which the derivative is being approximated is the

center of the points used for the approximation. The dependence of this derivative on the

values of at the points involved in the approximation is often represented by a “stencil”

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or “molecule,” shown in Figure 2.2.