1549301742-The_Theory_of_Difference_Schemes__Samarskii

Asymptotic stability 333

and the stability condition T < h^2 /2 also irnplies that T < 2 (D + 6.)-^1. In

the case of a symmetric scheme

h2

sin Irh

h

7r

and the condition T < h/7r is not burdensome. Thus emerged the typical

relation/= Tjh^2 < l/(7rh),sothat1<10/Jrforh = 0.1and1<100/Irfor

h = 0.01. The first eigenvalue A~ = D of the difference problem concerned

IS

where ~ = Jrh/2. Plain calculations show that

Ah= { 0.97 A^1

(^1) 0.9997 Ai
for h = 0.1,
for h = 0.01 ,
It is interesting to learn what may happen in dealing with the symmetric
scheme with O" = ~ and one of the possible steps T > T 0 , say T = m T 0 ,
m. > 1? Then max; I qk I is attained for k = N - 1:
Upon substituting here
1
T 6_
1
we get p= e-r.S+r~(r,.S) with
41n^2 '
1-2T-^1 6.-^1
1+2T-^1 6.-^1
D 2
4m ../[Ii
TD
4m^2

• TD

Thus, in the symmetric scheme with O"
revealed for m > 1:

~ the improper asymptotics is