1549301742-The_Theory_of_Difference_Schemes__Samarskii

442 Stability Theory of Difference Schemes

where

Proof In what follows we distinguish two separate tasks.

~ lj Stabili!J with respect to the initial data. Since the conditions

R > A/4 and B > 0 of Theorem 1 are fulfilled, we have for a solution of

problem (65) with <p = 0 the esti1nate

II Y(t + r) II< II Y(t') II, t' < t)

and, in particular,

(71) II Y(t + r) II< II Y(r) II,

where II Y(t + r) II is specified by formula (70) being a particular case of

formula (24) with A= E and R = ~(cr 1 + cr 2 )E incorporated.

2) Stability with respect to the right-hand side. Consider problem

(63) for y(O) = y(r) = 0 and seek its solution in the form

(72)

n

Yn+l = L T gn+l,s >

s=I

Yo= 0,

where gn+ l,s as a function of n for fixed s = 1, 2, ... , n satisfies equation

(63) with <p = 0 for n > s + 1 and the initial data

(73) gs+l,s + 2 0"1 TA gs+l,s = 2 gs ' s = 0 ·

Substituting (72) into (63) and taking into account (78), we conclude
that (72) is just the solution of problem (63). As we stated in (71), on the
strength of stability with respect to the initial data we have for gn s
'
(7 4) II Gn+l,s II <II Gs+l,s II for fixed s = 1, 2,... ,

where 11 Gn+l,s II is expressed through gn,s and gn+l by relation (70). We
find from (73) that'gs+l,s = 2(E+ 2cr 1 rA)-^1 <p and establish the relations
II E + 2cr 1 r A)-^1 II< 1 and II gs+l,s II < 211<p 8 11- This is due to the fact that
E + 2cr 1 rA > E for cr 1 > 0
By assumption, g 8 s = 0. This provides support for the view that
'
2 1 2 1 ( 1) 2
II Gs+l,s II = 411 gs+l,s II + 2 0"1 + 0"2 - 2 II gs+l,s II

= ~ (cr1 + cr2) II gs+l,s 11

2

< 2 (cr1 + cr2) II <fs 112,