1549301742-The_Theory_of_Difference_Schemes__Samarskii

Asyn1ptotic stability 337

for which

I f (μ) I - f (μ1) < o for if P1 < 1 ·

It is necessary to consider merely the case (1-0") μ > 1, since otherwise

I f (μ) I = f (μ) < f (μ 1 ) for μ > μ 1. In that case the difference in question

becomes

(1-0")μ-l

(1+~(}μ)2

= D^1 { 2 + ( 2 (} - 1) μ^1??

1 + 4" (}-μ;

• ( ( 1 - 2 (}) + 2 (} ( 1 - (}) μ l + ~ 0"^2 ( 1 - (}) μ; ) μ

where D = ( 1 + ~ O" μ)

2

( 1 + ~ O" μ 1 )

2

.

Recalling that O" = 2 - V'J,, 0"^2 = 4 O" - 2 and μ 1 < 1 we get the

lower estimate which will be the subject of special investigations in the

near future:

> D^1 { 2- ( 20"(1-0")+^1 ?( )^1 3 ?}

4

(}- 1-0")-(20"-1) μ+

4

(} μ-

= ~ { 2 - (6.5 - 10.5 (}) μ + (3.5 (} - 2) μ^2 }.

The square trinomlal in the curly brackets equals 0.05 μ^2 - 0.349 μ + 2 and
has the negative discriminant d = 0.349^2 - 0.402 < 0. Because of this, we
deduce that f(p 1 ) - If(μ) I > 0, rneaning

under the constraint μ 1 = T 15 < 1, which is valid if being replaced by the
condition TA 1 = T 7r^2 < 1.