1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable three-layer schemes 429

(A, B and Rare, in general, variables, that is, they depend on tn)· From

such reasoning it seems clear that problem (1) is solvable if an inverse

operator (B + 2r R)-^1 exists. In the sequel this condition is supposed to be

satisfied. Moreover, we take for granted that

(3) the operator B+2rR is positive definite.

Further development of the three-layer scheme (3) is connected with

the functional known as the compound-norm:

(4)

where 11 · ll(li) and II · 11(1 2 ) are suitable nonns on the linear syste1n H.

In order to understand the structure of this norm a little better, it is ap-

propriate to introduce the space H^2 = H Efl H being the direct sum of two

copies of H. The space H^2 is defined as the set of all vectors of the form

y(o:) EH, CY= 1, 2,

where the operations of addition of vectors and multiplication of a vector

by a number are carried out in a coordinate-wise fashion:

y + y = {y(l) + y(l), y(2) + y(2)})

The norm on the space H^2 is natural to be defined by

In our case the vector

possesses the coordinates Yl = ~(Yn+l +Yn) and Y 1 = Yn+l -yn- It is
easy to see that functional ( 4) satisfies all the axioms of the norm, namely
II aYn+1 II = lal 11 Yn+l II, II Yn+l II > 0 for any Yn E H, Yn+l E H and
II Yn+1 II= 0 only for Yn = Yn+l = O; II Yn+I + Yn+l II< II Yn+1 II+ II Yn+l II·
We now in a position to define the notion of stability of scheme ( 1).
The three-layer scheme ( 1) is said to be stable if there exists the norm ( 4)
and for all sufficiently small T < r 0 and lhl < h 0 one can point out positive
constants ]\1[ 1 and 1v1 2 independent of r, h and disregarding to the choice