¦ ¦© "

¦ ©

¦ "

¦ ©

¦

V V V

6

0

In the particular case where the mesh sizes and are the same and equal to a mesh

size , the approximation becomes

— 0 B 1 6

¡ —C —

—C

BV 6 0 ¦B V 6 ©© 0 B V

6¦ ¦

3

¦ "

¦

¦

¦

V

C C Y

t

©0 B — 3C© 0

¢

6 BV 6 ¨ (¦ &¥

3 3

"

¦ ¦© ¦ ¦©

V

C

which is called the ¬ve-point centered approximation to the Laplacean. The stencil of this

¬nite difference approximation is illustrated in (a) of Figure 2.3.

” ” ”n

uu u ™

n

£ ¢

¤§ ¥ §¢

¡

¡© £ £ ¥ £ ¤ § ¨¡ ¢

§¥ © ©

(a) (b)

1 1 1

1 -4 1 -4

1 1 1

¥ ¡¢

D ©T

Five-point stencils for the centered difference ap-

proximation to the Laplacean operator: (a) the standard stencil,

(b) the skewed stencil.

0 ¦B V 6

¨ ¨

©

¦

Another approximation may be obtained by exploiting the four points

0 C

BV 6

¦ ¦©

located on the two diagonal lines from . These points can be used in the same

C

manner as in the previous approximation except that the mesh size has changed. The cor-

responding stencil is illustrated in (b) of Figure 2.3.

(c) (d)

1 1 1 1 4 1

1 -8 1 4 -20 4

1 1 1 1 4 1

¥ ¡¢

D T

Two nine-point centered difference stencils for the

Laplacean operator.

The approximation (2.17) is second order accurate and the error takes the form

6 ¢ ¢

e C —

B¡

V V

—

0¢ 6¢

¦ ¦

There are other schemes that utilize nine-point formulas as opposed to ¬ve-point formu-

las. Two such schemes obtained by combining the standard and skewed stencils described

above are shown in Figure 2.4. Both approximations (c) and (d) are second order accurate.

However, (d) is sixth order for harmonic functions, i.e., functions whose Laplacean is zero.

£

”’©§ d¥ u„£

un u n

¢ ¢ £ § © ¥ ©¨¤© £ £

¡ ©¢¥ ¡

¤ ¥AB¥A £ ¡§)53 ¦ ¨ ¦

¢ ¨ 1¤¢¥ ©

6 £§

¡¡ I¨ ¨ 3 ¤&¨ #B 3

§ 3©

Consider the one-dimensional equation,

‘t

¡ )C z B

B V3 B ¨ 2 ¦ &¥

7

" ¦¦

¦ ¦ ¦