1549301742-The_Theory_of_Difference_Schemes__Samarskii

Asymptotic stability 329

This stage of the process refers to the regular behavior. When the so-

lution of a difference scheme for problem (1) also possesses the properties

similar to (2) and (3), the scheme is said to be asymptotically stable.

We now deal with the scheme with weights

Yt = A ( (} f) + ( 1 - (}) y ) , t=jr>O,

(4) Yo= YN = 0' y(x, 0) = u 0 (x),

Ay = Yxx ·

The solution of the problem under consideration has been obtained in Sec-

tion 1.4 by the method of separation of variables and can be expressed by

the following series

where

N-l

y/ = y(x;, tj) = L ck qi Xk(x;),

1-(1-(J)TA~

l+(JTA~

k=l

/\k \h --^4

h2

7fhk

sin^2

2

N-l

(y,v)= L Yivih.

i=l

The orthonormality of the system { Xk} gives

N-l N-l

II uj 112 = 2= < p2j L c~ = p2j^11 yo 112,

k=l k=l

which assures us of the validity of the estimate

(5) p = l<k<N-l lnax I qk I.

Scheme ( 4) is said to be p-stable with respect to the initial data if estimate
(5) is valid. An estimate of the form (3) is admissible for the solution of
problem ( 4) when p < 1.