1549301742-The_Theory_of_Difference_Schemes__Samarskii

Iterative alternating direction rnethods 721

In order to understand the nature of this a little better, we refer to a

model problem posed in Section 2 all over again for which

c5 1 = c5 2 = c5 = 4 sm. 2(7rh)/ T h" ") ,

f::..1 = f::..2 = f::.. = 4 cos 2(7rh)/ 2 h^2 ,

~ Irh

..fij = v tg~ 2 ~ 2 ~ 1. 5 7 h.

The n1ain goal of subsequent considerations is the con1parison between

ADM of the type ( 17) with para1neters ( 25) and the explicit 1nethod with

optimal set of Chebyshev's para1neters

Yk+1 ----+Ayk - Yk =f, k=O,l, ... ,

arising in Section 2 and requiring no less than n l^0 J ( c) iterations:

n(o)(c) = ln(2/c)

2 VE, '

In that case 11 = c5 1 + c5 2 = 2 c5, 12 = f::.. 1 + f::.. 2 = 2 f::.. and~ = c5/ f::.. T).

Thus, we 1night have

n(o)(c) ~ ln(2/c) ,

2-ftj

-r2 ll) (c ) ~ ln(l/c) i

4-ftj

thereby justifying that nl^0 l(c) ~ 2 nl^1 l(c).

vVhen solving the model problem concerned, the transition from the

kth iteration to the ( k + 1 )th iteration is perfonned either in 9 steps or in

26 steps: 5 operations of addition and 4 operations of multiplication during

the course of the explicit Chebyshev 's n1ethod and 12 operations of addition

and 14 operations of multiplication in the case of ADM in connection with

the double elimination (first, along the rows and then along the colmnns).

This provides reason enough to conclude that in the case of noncommutative

operators the first method is rather economical than the second one. Both

1nethods require 0( ! In~) iterations in the process of solving the model
h c
problem under consideration.