1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable three-layer sche1nes 437

Lemma 2 If y(O) = y( r) = 0, then

t

(42) II y(t) 112 +II y(t + r) 112 < 4t LT II Yr(t') 112 -

t'=r

Indeed,

t

y(t + r) + y(t) = 2 L ryr(t'),

( 43) t'=r

lly(t+r)+y(t)ll^2 <4t L rl1Yr(t')ll^2 -

t'=r

In the notation wn = Yn - Yn-l we obtain

w I -- (^0) '
which implies the inequality
II Wn^1 +1 II < 2 T II Yt,n' II+ II Wn^1 II, n' = 1,2, ... ,n.
Summing up the preceding over n' from 1 to n we find that
n
II Wn+1 II < 2 L T II Yt,n' II
n'=l
or
t
II y(t + r) - y(t) II < 2 L r II Yr(t') II,
( 44) t'=r
II y(t + r) - y(t) 112 < 4t L T II Yi(t') 112-
t'=r
Putting inequalities ( 43) and ( 44) together with the obvious identity
( 45) 11 y( t + T) + y( t) 112 +11 y( t + T) - y( t) 112 = 2 (I I y( t) 112 + 11 y(t + T) 112) )
we arrive at ( 42). Substituting estimate ( 42) into ( 41) yields (38).