1549301742-The_Theory_of_Difference_Schemes__Samarskii

722 Methods for Solving Grid Equations

5. Factorized iteration sche111es and ADM. The main idea behind this ap-
   proach is connected with the equivalence between ADJ!I from the preceding
   sections and the two-layer iteration schen1e

(26) B Yk+i - Yk + Ayk = f

T

with the factorized operator

(27)

and the iteration parameter

(28)

To make sure of it, we rewrite ( 17) as

where B 1 = E+w 1 Ai, B2 = E+w 2 A2, C\ = E-w 2 A1 and C2 = E-w 1 A2.

The elimination of the iteration Yk+l/ 2 can be done by successively applying

the operator C 1 to the first equation and the operator B 2 to the second one.

Combination of the resulting equations with further reference to the useful

relations Bi Ci = C 1 B1, w 1 Ci+ w 2 B1 = w 1 (E - w 2 A1) + w 2 (E + w 1 A1) =

(w 1 + wJE gives

(30)

It seems worthwhile to reduce equation (30) to the canonical form (26) by
observing that BiB2 - C1C2 = (w 1 + w 2 ) (Ai + A2)· The arguments in
reverse order are obvious.
Let us stress here that the applications of the above framework to
nonc01nmutative operators Ai and A2 could result in wrong reasoning in
light of the property that operator (27) is non-self-adjoint and sche1ne (26)
does not fall within the category of two-layer iteration sche111es lying in the
fundamentals of the general theory.
Before going further in more detail on this point, it is worth mention-
ing that the factorized operator (27) is self-adjoint and positive: B = B* >

0. To decide for yourself whether the obtained results are acceptable for
   comnrntative operators Ai and A2 and conditions (5)-(6), the first step is
   to discover from (30) the structure of the transition operator of sche111e ( 26)
   such as
   L.-10' ,, - B-lr• a Va' o:=l,2.