1549301742-The_Theory_of_Difference_Schemes__Samarskii

Two-layer iteration schemes 663

All this enables us to write clown

To

Tk = '

1 +Po tk

(29) k= 1,2, ... ,n,

thereby completing the task of motivating the choice of the para1neters

T 1 , T 2 , ... , Tw The expression for qn = 1/ITn(to)I, t 0 = -1/p 0 , is needed as

further developments occur. Since 1tl 0 > 1, applying formula (25) to Tn(t 0 )

yields

By 111inor changes the expressions in various brackets are modified into

1 ~ l+Jl-p6 1 ( ~)

Po + V P! - l = Po = Po l + V ~

~ -v 1 1 = 1 + ~ - 2 y/[, = 1 - y/[, = P1

Po P6 1 - ~ 1 + -./[

and, hence,

(30) IJn =

2 p:'
1 + p~"
Thus, for scheme ( 14) with optimal set of the parameters r 1 , r 2 , ... , Tn
given by formula (29) the relation

( .31)

is valid wit.h q 11 st.ill subject to (30).
The number n = n(t:) is so chosen as !.o satisfy the condition
2pn
qn = I < E
1 + p~n -.

For this, it suffices to require in the ful'ther derivation that P7 < t:/2 or,
what a111ounts to the same, that

(32)

ln ( 2 c)

n>---

ln ( 1/ p 1 ) •