w "

( (

b b

© ©

on a rectangular domain with Dirichlet boundary conditions. The equations are discretized

¤ ©

with centered ¬nite differences using points in the direction and points in ˜w ˜w

!

yq t 5 ¢ | v§¦ ¥ t¦C£ ¡r5© £

„ 5| 5 §|

¡¨ ¡C

C

!

b

the direction, This results in the system of equations

a± ¯ i ¯®

¡ 4¬

¨

£ £

w –

(

¡

in which the matrices and represent the three-point central difference approximations

to the operators T T

Xb X

c ( b b"

© ©¨

and

( (

b

© ©

respectively. In what follows, the same notation is used to represent the discretized version

of the unknown function . £

b

©

The ADI algorithm consists of iterating by solving (4.49) in the and directions

alternatively as follows.

˜{

“

˜h¤ v ¢

¡¦ ¥©4H5 £R¥¤B9

¤5E@ ¢ 3 A ¢ ¥ 8CD ¨¥7@

§¦E

©

6@

0 (%&$

'#

) 2

1

T T

¬

1. For until convergence Do:

V( I¡( ¡Y

(

W

£X X

yT T

¬ ² ¨w

2. Solve: £

i H w w

H

¡

UQH £ X i H £ X

¬ ² ¨

3. Solve: w H

"VH H

WU QH

U

4. EndDo

( H

¬

Here, , is a sequence of positive acceleration parameters.

¡™(

(˜

W

y

The speci¬c case where is chosen to be a constant deserves particular attention.

H

In this case, we can formulate the above iteration in the usual form of (4.28) with T T T T

a± — i ¯®

¡ ¡

X X X X

¬ ² ²X

w T T W “ T w

¦ (

W“

R±`— i ¯®

°

¡ ¡

X X

¬ ² ² ¨ W “

w w

W “

or, when , in the form (4.22) with

Y

T T T T

a±0— i ¯®

¡ ¡

w X X X X

¬ ¬ ² ²

w

(

y y

˜ ˜

Note that (4.51) can be rewritten in a simpler form; see Exercise 5.

The ADI algorithm is often formulated for solving the time-dependent Partial Differ-

ential Equation T T

a±a— i ¯®

£ £ £

Xb Xb

c

©¨ b T © © © ¬ w

¥

T ( "

(

b

T

X (Y ¥ X ¡Y !"& ¡Y $ ¥ ¡ X ( b ©

& ( $Y ¥

on the domain . The initial and boundary

(

(

(

y

y

conditions are: T

T

T

X $ b (T X b $ %T ± ¯ — i ¯®

X $ b %T