1549301742-The_Theory_of_Difference_Schemes__Samarskii

738

where

Methods for Solving Grid Equations

1-~

Po= l + ~,

l, c -- /1 ,

/2

so that T = T 0 B. The second sununand on the right-hand side of (36) will

be the subject of special investigations. The outcome of this is

11 ( 1 - B) y - T Ai y 112 = ( 1 - B)^2 II y 112 - 2 T ( 1 - B) (Ai y, y) + T^2 II Ai y 112

= (1 - B)^2 IIy11^2 + T^2 II A1y11^2

< [(l-!1)^2 +T"1;J · llYll^2

[(1-B)^2 +T 02 B^2 1;J · llYll^2 ,

which assures us of the validity of the following inequalities for T = T 0 B:

II s II:::; f(B),

(38)

f(B) =(}Po+ j(l - B)2 + g2 a2, a^2 =To^2 /3^2.

In order to find the mini1num of the function f ( B), we calculate and analyze
its derivative

l-B-a^2 B O'-a^2

f'(B) =Po - j(l - ())2 + a2 g2 =Po - Ja2 + a2' <Y =

1 - (}

(}

The equation f' (0) = 0 gives p 0 J ct^2 + a^2 = ct - a^2. It may be viewed as a
quadratic equation with respect. to ct:

( 1 - p~) a^2 - 2 a^2 a+ a^2 ( a^2 - PZ) = 0

with the first root
a+ Po jl - p~ + ct^2
et= a 2
1 - Po

We note in passing that the second root is unacceptable in connection with
the possible negativity for s0111e value of the para111eters a and p 0. Also, it
will be sensible to introduce the notation x = 13 / j1 1 / 2 +1;, so that

,; =

x2

ct^2 = T; 1;

411 /2 x2 (1 - PZ) x2

l 2 fl /2 ,

-x (11 +12)^2 1-x^2 1- x^2

a2^1 - Po^2

x2 1-x 2.