1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternative-triangular 1nethod^705

Cumbersome calculations for a proper choice of d(x) will be excluded

from further con.sideration. It .seem.s worthwhile giving only the final results

for the later use

(84) d(x) == I: Q + - _Q'. - _Q'.

2

( a+ 1 I a+ a 1) 1

n=l fjn ht~ 2 fjn ht h-;; ~ + V'n ,

(85)

/3=3-et, Ci= 1,2.

The functions v;n\x) and v~n\J:), n = 1, 2, are declared to be solutions of

the relevant problems

(86)

(87)

(n) = _l_I at - an I·

p^2 211 o, h+ O' h-O'

vCn)I 2 /cv = (^0) '
These will be given special investigation by means of the elimination method
with 0( 1) operations required at every grid node. With the intervention
of four new functions v;n)(x), v~n)(x) it is not difficult to develop the grid
functions <f!n(xf3) and t/Jn(;t:(J) of one variable and then specify d(x) by for-
mula (84). Under such a choice we obtain
(88) c5 = 1 '
leaving ns with the iteration parameters w 0 and { r 1 J and the iteration
scheme
k+l k
(D+wA1)D-^1 (D+wA2) y -y +At =<p, k=O,l, ....
Tk+l