1549301742-The_Theory_of_Difference_Schemes__Samarskii

696 Methods for Solving Grid Equations

The constants f; and ~ remain as yet unknown. Since f; 1s the smallest

eigenvalue, it is not difficult to show that

By virtue of the relations

plain calculations permit one to conclude that the constant ~ is equal to

p 4

2..:= h 2 • vVith knowledge of b, ~, c 1 and c 2 simple algebra gives w, n(c)

a=l °'

and { r; }, which constitute what is needed in the applications of ATM. The

total number of the iterations

(56) n () E > n 0 () E -_ /f;-^2 ln(2/c) r.:; ,

• c 1 2 v 2 !/ii

is proportional to V /9i.. c1
In the case of a cubic grid ( h 1 = h 2
p-dimensional cube we might have

hp = h) m a unit

4p. 27rh 4p. 27rh

f; = -

1

i~ ,, sm - 2 , ~ = h'.' - , 'f) = sm - 2 ,

thereby justifying that the total number of iterations is independent of the

number of observations.

Tl le iteration.. k+l y. is recoverec l f rom t e h equations.

(57)

( 58)

k k k - k k

F =By - T1.:+ 1 (Ay - <p) =By + T1c+i (Av + <p).

A solution of problem (57) is given by kt' = "t
1
for x E wh and by t = μ
c ior x E ih, so t h at k+1 v - y = k+l y - y. Th e precec l' mg d'.cr tHerence operators

B 1 and B 2 can be written as

(59) (

p l) p 1

B 1 y = y + w R 1 y = 1 + w 2..:= h 2 y - w 2..:= h 2 yC -^1 " l ,

Cl'=l Q' l.-Y=l^0