tY

z CC PB C

C

#B V ¨ ¢¦ &¥

7 a

7

V

U˜

—

The interval [0,1] can be discretized uniformly by taking the points

˜ i1)1z @@

— a

7

¦

C — ˜ B B

(

& "B V

¦

where . Because of the Dirichlet boundary conditions, the values

C

4 0

"

¦

and are known. At every other point, an approximation is sought for the exact

V V

C C B ‘

¦

solution . V

If the centered difference approximation (2.12) is used, then by the equation (2.18)

4 0 0

¦

expressed at the point , the unknowns satisfy the relation V V V

&

¡ 6 0 4 — 0

V3 V3

V

&

C B ¡ ¡ ˜ D D

¦

in which . Notice that for and , the equation will involve and

&V

˜ £

Y

04 V which are known quantities, both equal to zero in this case. Thus, for , the

‘

linear system obtained is of the form

h

¡

¦

where

3444

99 8 3

44

99 3 3

9 3 3

6

3 3

5

A 3 3

3

¦ 4¨I ¦

¥B¥A

¡ A¡ ©B ¨6 %

Consider now the one-dimensional version of the convection-diffusion equation (2.7) in

‚ ’¡

8

7

which the coef¬cients and are constant, and , using Dirichlet boundary conditions,

™— –

‚B z‘

£ 7 ¥

3 7

Y¥

t

¦

¦¦ ¦

V

`

V ¨

p £& &

¥B

V 8 #V 7

7

C C

In this particular case, it is easy to verify that the exact solution to the above equation is

given by

¤¥¤

3

BV ¦

C ¤¤

3

‚ ¥ (

where is the so-called P´ clet number de¬ned by

e . Now consider the approxi-

5 5

mate solution provided by using the centered difference schemes seen above, for both the

” ” ”n

uu u ™

n

£ ¢

¤§ ¥ §¢

¡

¡© £ £ ¥ £ ¤ § ¨¡ ¢

§¥ © ©

¬rst- and second order derivatives. The equation for unknown number becomes

p 0 — 6 l 0 4 0 0 4 V3 3

‚

V V V V

3 8

7

& &

5 –

(

or, de¬ning ,

Y¥

t

p 0 C — PB — 0 4 C •B

¨¦

3 3 3 7 &&

V V V

&

This is a second order homogeneous linear difference equation and the usual way to solve

…' …

it is to seek a general solution in the form . Substituting in (2.21), must satisfy

' V

z C — kB –' l 6 ' C ! PB

— W

3 3 7

0'

C •B C — PB 6 ' ( I

3

Therefore, is a root and the second root is . The general

solution of the above difference equation is now sought as a linear combination of the two