Operator-difference schen1es 385involve in further development the norms II · ll(lh)' II · ll( 2 h)' ... on the
space Bh, assuming II · llh to be a basic one on Bh.
At the next stage we introduce on the interval 0 < t < t 0 a T-step grid
WT = { t 1 = j T, j = 0, 1, ... , j 0 , T = t 0 / j 0 }, WT = { t 1 = j T, 0 < j < j 0 }
and pass to abstract Eh-valued functions y~~, 'P~t), etc. of one discrete
argument t = jr E wn so that y~~ E Bh far all t = jr E wT. Let A~t),
B~t}, c~~' ... be linear operators dependent on the parameters h and T
and acting from Bh into Bh for every t E wT. For the mon1ent, omitting the
subscripts h and T we can write down Yn = y(nr) = y(tn) = y, A(t), B(t)
and C(t). This should cause no confusion.
We call a family of difference equations of order r - 1
r-1
Bo(tn) Yn+l = L Cs(tn) Yn+l-s + fn' n = r - 2, r -1, r, ... ,
s=l
depending on the parameters h and T with operator coefficients Bo, cl, ... '
C,.-1, which are linear operators acting in the space Bh and dependent on
h and T, an r-layer operator-difference scheme or simply r-layer sche1ne.
If the inverse operator B-;i exists, a solution Yn+i of this problein can
be expressed in tern1s of the initial vectors y 0 , y 1 , ... and the right-hand
side f. As usual, we assume all the vectors to be given and consider only
two-layer and three-layer schemes
(1)(2)n=O,l, ... , y 0 given,
T 'Pn , y 0 and y 1 given.- The canonical form of two-layer schemes. Any two-layer scheme (1) can
be written in the form
(3)
n=O, l, ... , Yo E Bh given.
Indeed, by comparing (1) with (3) we see that B = B 0 and A
B 1 ) / T. Within more compact notations
Y = Yn = y(t,,), Y = Yn+i = y(tn+1) = y(tn + r),
y-y y-y
Y=Yn-1• Yt = T Yt = T