1549301742-The_Theory_of_Difference_Schemes__Samarskii

574 Economical Difference Schen1es for Multidimensional Proble1ns

In what follows the weighted scheme

(33)

y(O) =Yo, Y(r) = U1,

is treated as a primary one and it vvill be writ kn in canonical form for later

use:

(34)

where the unknown operator B + 2rR = E + 2o- 1 rA is sought. Let now

A = A 1 + A2. The factorized operator

will appear in place of ~he ope!::_ator B + 2r R as one possible way of con-

necting two operators B and R by a unique condition and it may be of

assistance in designing n1any factorized schemes. Later we will expound

certain exploratory devices for obtaining them. For example, this can be

done using ( 34) in the form

(B + 2 TR) 1ft + (B - 2 TR) Yt + 2 Ay = '). cp

and replacing the operator B + 2 TR by the factorized operator B -+ 2 TR, -

making it possible to write the scheme at hand in canonical form with the

aid of the well-known relations

Yt = yt + 0.5 TYft,

The outcome of this is

(35)

y(O) = Yo,

so that

Yt = ya t - 0.5 T Ytt •