1549301742-The_Theory_of_Difference_Schemes__Samarskii

430 Stability Theory of Difference Schemes

of Yo, y 1 , ip( t) such that for any Yo, Y1> ip( t) and all t = T, 2r, ... , ( n 0 - 1 )r

a solution of problem ( 1) satisfies one of the following estimates:

(5) llY(t + r)ll(l) <Mi llY(r)ll(lo)

• M2 O<t'<:;t max llip(t')ll( 2 ),

(6) llY(t + r)ll(l) <Mi llY(r)ll(lo)

• M2 r<t'<:;t max (llip(t')ll( 2 ) + ll'Pr(t^1 )ll( 2 )),

where II · 112 is some suitable norm on the space H, llY(t + r)ll(l) and

llY(r)ll(lo) are defined by the formula of the form (4), so that

(7)

1

llY(t + r)ll~l) = 4 llY(t + r) + y(t)ll~li) + llY(t + r) - y(t)ll~ 12 ),

(8) llY(r)ll(lo)^2 = 4 1 llY1 + Yoll(l~)^2 + llY1 - Yoll(l~),^2

where 11·11(1~) and 11 · ll(lg) are suitable nonns on the space H.

As far as constant operators A and Rare concerned, the norms llYll(l)

and llYll(l o) normally coincide. In the general case llY(t + r)ll(l) and

llip(t)11( 2 ) depend on t = nr, so that one should write llY(t+r)ll(l,t) instead

of llY(t + r)ll(l) and llip(t)ll( 2 ,t) instead of llip(t)ll( 2 ).
As we will see later, the norms II· ll(li) and II· ll(lz) are energy norms
constructed for the operators A and R. For this reason we will assume that
these operators are

(9)

( 10)

self-adjoint:

positive:

if H is a Hilbert space.

A-A* - ) R = R*,

R > 0,

2. The basic energy identity. We will carry out the derivation of the energy
   identity for the three-layer scheme ( 1) with variable operators A = A( t),
   B = B(t) and R = R(t). This identity is aimed at achieving a priori
   estimates expressing the stability of a scheme with respect to the initial
   data and right-hand side.
   By virtue of the relations