1549301742-The_Theory_of_Difference_Schemes__Samarskii

730 Methods for Solving Grid Equations

and the parameter T should be so chosen as to minimize the norn1 115'11. We

now know frorn Section 2 that this aim can be achieved by setting T = T 0 ,

so that the preceding becomes

(5)

allowing al tern a ti ve forms of writing:

( 1 + 0:) U = ( 1 + Cl') ,5' 1l + ( 1 + Cl') T 0 f ,

ll = ( 1 + ex) S' 1l - ct u + ( 1 + O:) T 0 f.

Having replaced (1 + o: )Sll by ( 1 + o: )Syk and O:ll by o:yk-l, we try to adapt

the explicit scheme, the parameter o: of which needs to be selected by the

approved rule in a minimal number of iterations. Unfortunately, more a

detailed exploration on this point and the convergence of scheme (3) are

not available in the present book. A final result can be obtained through

such an analysis by utilizing the fact that the residual rk = Ayk-f satisfies

the homogeneous equations

(6)

no nlatter how the initial value r 0 is chosen.

In a revised iilatenwnt of the problem for ct = p~ the e'itin1ate

(7)

is valid with

(8)

assunng

llAy,, -Jll < !/n llAYo -f II

1 2

q,, = P1 n (l + 1 - + P1 2 n ) ,

P1

n>

1 ( 1 - p

2

In - +In 1 + ~ n )

c 1 + P1

1

In -

P1

This is certainly so with

(9) n>

(^111) -^1 +I 11 (l + 2 1 /( c n)
c +"