1549301742-The_Theory_of_Difference_Schemes__Samarskii

Asymptotic stability 339

or

(^3 +2r>-h) k y)+i k -4Tj k +Tj-i k --^0 ,

whose solution is sought in the form Ti = qk. Omitting the subscript k for

a while, we investigate the quadratic equation related to qk:

(3 + 2 μ) q^2 - 4 q + 1 = 0' μ=TA,

with discriminant D = 4 (1 - 2μ). For D < 0 its roots q (l,^2 l are complex-

conjugate and q(llq(^2 ) = I q 12 = (3 + 2μ)-^1 , that is,

I

q(l, 2) I - 1 < 1 = I q 1 (1. 2) I ·

k - J3 + 2 T Ak J3 + 2 T Ai

Let now D > 0. In that case

ci,^2 )_ 2±y'l-2μ q(i)>O, q(2)>0,

q - 3 + 2μ '

A new function

VJ(μ)= 2+v'l-2μ

3 + 2μ

with the derivative

-^1. oVJ -^1
VJ 8μ (2 + y'l - 2μ) v'l - 2μ

q (2) = -----< 2 - y'l - 2μ^1.

3 + 2μ

2

< 0

3 + 2μ

is aimed at establishing that I q~^11 1 I < I q~^12 ) I for k 1 > k 2. Hence, max VJ(μ)

!'15.w'C,μN

is attained for μ = μ 1 = T \, giving

if 1 - 2μ1 > 0.

It may happen tha,t ~ Dk = 1 - 2μk < 0 for some k > 1, thus causing the
occurrence of the event that

Since (3 + 2μk)-^1 < VJ^2 (μ 1 ), the maximum value rnax I q~^1 •^2 ll, which can
k
always be attainable for k = 1, is equal to

p =

2 +JI - 2μ 1

if