( #¡ © £

( ¡ b © ( ( ( b © '¬ Y ( b ’© a±—— i ¯®

£

(& (

( ¡Y

(

where is the boundary of the unit square . The equations are discretized with respect

b ©

to the space variables and as before, resulting in a system of Ordinary Differential

Equations:

a± — i ¯®

£

¡ ¬ £ £ £

w

(

£

µ£ „ ¢

|5¥ j qz A C¡y 5

„ 5| | 5£§|

¡C ¢ ¡

C @¡

¨§ £ "!

¡

in which the matrices and have been de¬ned earlier. The Alternating Direction Im-

b

©

plicit algorithm advances the relation (4.56) forward in time alternately in the and

directions as follows: T T

¡

X X

² ¬

£ £

i w (

y y

¡˜

¢˜ H

QH

U

T T

¡

X X

² ¬

£ £

i

w

y y

¢˜

£˜

"QH

WU QH

U

The acceleration parameters of Algorithm 4.3 are replaced by a natural time-step.

H

Horizontal ordering Vertical ordering

19 20 21 22 23 24 4 8 12 16 20 24

13 14 15 16 17 18 3 7 11 15 19 23

7 8 9 10 11 12 2 6 10 14 18 22

1 2 3 4 5 6 1 5 9 13 17 21

AB@85¤4 2

9 76 3

The horizontal and vertical orderings for the un-

knowns in ADI.

©

Assuming that the mesh-points are ordered by lines in the -direction, then the ¬rst

¤

step of Algorithm 4.3 constitutes a set of independent tridiagonal linear systems of size !

each. However, the second step constitutes a large tridiagonal system whose three diagonals

²

are offset by , , and , respectively. This second system can also be rewritten as a set

Y

! !

¤

of independent tridiagonal systems of size each by reordering the grid points by lines, !

b

this time in the direction. The natural (horizontal) and vertical orderings are illustrated

in Figure 4.6. Whenever moving from one half step of ADI to the next, we must implicitly

¥¤

work with the transpose of the matrix representing the solution on the grid points. !

This data operation may be an expensive task on parallel machines and often it is cited as

one of the drawbacks of Alternating Direction Methods in this case.

ADI methods were extensively studied in the 1950s and 1960s for the particular case

¡

of positive de¬nite systems. For such systems, and have real eigenvalues and the

¡

following is a summary of the main results in this situation. First, when and are

¬

Symmetric Positive De¬nite, then the stationary iteration ( , for all ) converges. Y

W

H

For the model problem, the asymptotic rate of convergence of the stationary ADI iteration

#

using the optimal is the same as that of SSOR using the optimal . However, each ADI

step is more expensive than one SSOR step. One of the more important results in the

ADI theory is that the rate of convergence of ADI can be increased appreciably by using

a cyclic sequence of parameters, . A theory for selecting the best sequence of ™s is

H¡ T H

well understood in the case when and commute [26]. For the model problem, the £ X

§¥¤2 ¤

¤¦

parameters can be selected so that the time complexity is reduced to , for

details see [162].

³ ¡C

y £ w ¢ ¡y ¡|

|5 | §| C

vcQ—tc™£™

—™ “ ¢

1 Consider an tridiagonal matrix of the form

§¦¤

¤¥

a± — i ¯®

„

¨

©

‚