1549301742-The_Theory_of_Difference_Schemes__Samarskii

Operator-difference sche111es 389

Inequality (12) expresses the property of continuous dependence which

is uniform in h and T of the Cauchy problem ( 4) upon the input data. Here

and below the ineaning of this property is stability. A difference scheme

is said to be absolutely stable if it is stable for any T and h (not only for

all sufficiently s1nall ones). It is fairly common to distinguish the notion

of stability with respect to the initial data and that with respect to the

right-hand side. Sche1ne ( 4) is said to be stable with respect to the

initial data if a solution to the hmnogeneous equation

( 4a) Byt + Ay = 01 t=nr>O, y(O) = Yo ,

satisfies the estimate

(12a)

Scheme (4) is said to be stable with respect to the right-hand side if

for a solution to equation (4) with the zero initial conditon y(O) = 0

( 4b) B Yt + Ay = <p, y(O) = (^0 1)
the inequality holds:
(12b) II Yhr(t + r) llcih) < (^1112) O<t'<t inax II <fhr(t') llc (^2) h ) ·
For later use, we approve for a solution y of problem ( 4) the decmnposition
y = yCI l + yC^2 l, where yCI l is a solution of problem ( 4a) and yC^2 l is a solution
of problem (4b). On the strength of the triangular inequality
the combination of estimates (12a) and (12b) gives estimate (12). In a
similar way it makes sense to introduce the notion of stability for three-
layer schemes. However, in this case we are to consider the pair of vectors
Yn+l = { Yn, Yn+l} with the norm of the special type
where II 11(1~) and II · llci~·) are suitable norms on the space B1i. Various
norms of the type (13) appear, time and again, in the stability analysis of
three-layer schemes by ineans of the energy inequality method (for more
detail see Chapter 5, Section 6).