X

3

¥ ¡¢ T

D

The three-point stencil for the centered difference

approximation to the second order derivative.

The approximation (2.8) for the ¬rst derivative is forward rather than centered. Also,

3

a backward formula can be used which consists of replacing with in (2.8). The two

formulas can also be averaged to obtain the centered difference formula:

—

BV9 BV BV 3 3

C C

"

¦ "

¦ "

¦

Y

t

¨ ¢ ¦ &¥

C ¦9

”’©§ d¥ u„£

un u n

¢ ¢ £ § © ¥ ©¨¤© £ £

¡ ©¢¥ ¡

¡

It is easy to show that the above centered difference formula is of the second order, 4

&

while (2.8) is only ¬rst order accurate. Denoted by and , the forward and backward

difference operators are de¬ned by

B V 4 Y

t

C ¨ £¦ &¥

—

BV BV 3

"

¦ "

¦ "

¦

tY

C CC

C ¨ ¤¦ &¥

BV& BV BV 3 3

"

¦ "

¦ ¦

C

All previous approximations can be rewritten using these operators.

In addition to standard ¬rst order and second order derivatives, it is sometimes neces-

sary to approximate the second order operator

9 9

‚ B

"

¦

9 9C

¦ ¦

A centered difference formula for this, which has second order accuracy, is given by

‚¡ 6 — £V&

9 9 6B¡

4

Y

t

‚ V

B ¨ ¦ &¥

6 0

¦

9 9C

¦ ¦ & C

6 C C0

4 B 6 0 4 ‚ ‚ B 6 0

V3 3 V3

0

V V

&

&

¦ ¦

B¥AB¥A

A¡ ¡ G' ¨ I' 0' &¨#% !5& ¨ 6¨% ¨ I¨ §¨ 3 ¤©

3

§3

1 1

!# ' §¨

§

§§ §§

If the approximation (2.12) is used for both the and terms in the Laplacean oper- 3 3

6 ©¦

6

0 0

¦

ator, using a mesh size of for the variable and for the variable, the following 3 3

3

second order accurate approximation results:

0 B — C 6 06 B l C 6 0 — 0

¡ BV 6

3 3

¦ ©

¦ ¦ ©

¦ ¦ ©

¦

V V

—C

BV

¦

0

C

C 6 6 0 B — C 6 66 0 B C 6 — © 0 B

6 3 W

3