¡

the assumptions will be simpli¬ed slightly by taking a vector function that is linear with

respect to . Speci¬cally, assume

V

¤

¤ 0@

V

¡

V@

V6@

Y

¤@

¤ ‚ C B¤

¡ ¡

Note that, in this case, the term in (2.37) becomes . In addition, the V V

right-hand side and the ¬rst term in the left-hand side of (2.38) can be approximated as

follows:

¢ 1

¢ ¢

0R¥

V V

9¢ 9 )

¦ ¦

R¥

£

0

)

Here, represents the volume of , and is some average value of in the cell .

These are crude approximations but they serve the purpose of illustrating the scheme.

The ¬nite volume equation (2.38) yields

— ¢ ) 9 ¤ ˜

¥¢

¤@ Y

t

V ¨ ¢ ¢ &¥

¤

)

V

R

The contour integral

¥¢

¤˜ 9

V ¤

R

is the sum of the integrals over all edges of the control volume. Let the value of on each V

…Q …

edge be approximated by some “average” . In addition, denotes the length of each V ¤

edge and a common notation is

… ¤ ˜ … … ¤

¤ ¤

Then the contour integral is approximated by

¥¢

¤ … ¤˜ ¤ … Q … ¤ ¤ … Q

¤˜ & &

… Y

t

¨ £ £ &¥

9

@ V V @ V @

¤ ¤ ¤

R ¡¢ ¡¢

£ £

The situation in the case where the control volume is a simple triangle is depicted in Figure

2.12. The unknowns are the approximations of the function associated with each cell. V V

These can be viewed as approximations of at the centers of gravity of each cell . This V

type of model is called cell-centered ¬nite volume approximations. Other techniques based

on using approximations on the vertices of the cells are known as cell-vertex ¬nite volume

techniques.

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