1549301742-The_Theory_of_Difference_Schemes__Samarskii

330 Difference Schemes with Constant Coefficients

2. Asymptotic stability. Let us describe rathel' mild conditions under which
   the behavior of the solution y j of the difference probletn ( 4) becomes regular
   if one makes tj = jr large enough:

(6)

Clearly, it is possible only in the case when the first harmonic dominates,

that is, max I qk I is attained for k = 1, so that

k

(7) max qk = p =

l<k<N-l

1-(1-cr)rA~

1 +(TT A~

What is available to find out the conditions ensuring the validity of formula

(7) is the function

.f(μ)= (l-(l-cr)μ^1 )

2

-(l-(1-cr)μ)

2

,

1+crμ 1 l+crμ

where μ 1 = TA~ and μ = T AZ > μ 1 under the natural premise .f(μ) > 0,
μ > μ 1 • Alternative forms of the function f(~t) may be chosen in a number
of different ways. In subsequent reasonings we prefer one of them:

f{ = ( 1 + (T μ1 )^2 ( 1 + (T μ )^2 > 0 ,

f 2 (μ)= (l-(1-cr)μ 1 ) (l+crμ)+(l-(1-cr)μ) (1+crμ 1 )

= 2 + ( 2 (T - 1 ) ( μ + μ1) - 2 ( 1 - (T) (T μ μ1 ,

implying for T < f 0 that

if (T -- (^1) ,
if (T = 0 ,
where f. o = 2 (c5 + b.)-^1 ' c5 = Ah 1 and ,6. = Ah N-1. In particular, we have