388 Stability Theory of Difference SchemesThe passage from (6) or (8) to (2) leads to the equation for determining
the value Yn+l on the upper layer t = tn+l:( B + 2 TR) Yn+l = 2 T ( 2 R - A) Yn + ( B - 2 TR) Yn-1 + 2 T 'Pn ,
thereby clarifying that problem (8) is solvable if the inverse operator ( B +
2 TR )-
1
exists. Moreover, the value Yn+l can be expressed in terms of
the values Yn and Yn-l on the preceding two layers. Therefore, two initial
vectors Yo and y 1 (or Yo = y(O) and y 0 = Yt(O)) are required to be given
in such matters. If B 0 = B + 2 TR= E is the identity operator, then the
three-layer scheme (8) is said to be explicit; we thus have for itYn+l = 2r(2R-A)Yn+(B-2rR)Yn-1+2r'Pn·
In the case when B + 2 TR# E, scheme (8) is said to be i1nplicit.
Sometimes it may be useful to have at our disposal together with the
canonical form (8) the three-layer scheme in the form (2) or(11) B yo + ( E + r^2 R) Yt t + A Y = 'P ·
tThis equation is obtained frorn equation (8) by the formal substitution of
E/r^2 + R for R.- The notion of stability. The notion of stability for three-layer schemes
is of our initial concern. By a two-layer scheme we mean a set of operator-
difference equations ( 4) depending on the parameters h and T. We pre-
assumed here that the operators A and B are given on the entire space
Bh·
As a matter of fact, we will consider the set of solutions { Yh 7 (t)}
of Cauchy problem ( 4) dependent on the input data { 'Pi 17 (t)} and {Yoh}·
Scheme ( 4) is said to be well-posed if for all sufficiently small T < r 0 and
I h I< ho
(1) a solution of problem ( 4) exists and is unique for any initial data Yoh E
Bh and right-hand sides 'PhT(t) E Bh for all t E wT;
(2) there are positive constants M 1 and M 2 independent of h and T and
disregarding to the choice of Yoh E Bh and 'Ph 7 (t) E Bh such that for any
t E w 7 for a solution of problem ( 4) the estimate holds:
where II · ll(ld' 11 · 11(1~) and 11 · 11( 2 ") are suitable norms on the space Bh.