1549301742-The_Theory_of_Difference_Schemes__Samarskii

716 Methods for Solving Grid Equations

and Ct= /3 = 1 for .\(Ai)= ~ 1 , .\(A2) = ~ 2 , the constants p, q, r, T) are

found by the formulas

( 14)

l-t

T)=l+t' t=

(~1 - c51) (~2 - c52)

(~1 + c52) (~2 + c51),

(15)

r= --------~ ~1 - ~2 + (~1 + ~2) p

with x > t and p > 0.

l-p

q = r+--

~1

At the next stage, with reasonable accuracy c > 0 and knowledge of

the spectral bounds be,, ~CY of the operators ACY simple algebra gives TJ, p, q,

r by formulas (14)-(15). When providing a prescribed accuracy c > 0, that

is, II Yn - u II < E II Yo - u 11, it is necessary to perform n = n(c) iterations

that can be most readily evaluated by the approximate formula

(16)

1 4 4

n(c) ~ - In - In -.

7f2 [ 17

In the new notations

2 j - 1

(} -- ~ 16 T) 2 (1 + ~ 2 T) 2) , (]' = j = 1,2, ... ,n,

2 1)

the quantities wj are specified by the formulas

w. - ---------(1+2G)(l+G")

) - 2eaf2(l+e1-a+e1+a)' j=l,2,. .. ,n.

In agreement with (10) the undetermined parameters become

(l)_qwj+r

T· J - l + wj p '

making it possible to solve problem (7).
It should be noted that for particular values c5 1 = c52 = c5 and ~ 1 =
~2 = ~ formulas (14)-(15) give x = ~, p = r = 0, q = 1/ ~ and ~ =
(1-17)/(l + TJ), 17 = c5/~. Via transforms (9)-(10), amounting for now to