1549301742-The_Theory_of_Difference_Schemes__Samarskii

726 Methods for Solving Grid Equations

helps clarify what is clone. As a matter of fact, searching for ~h+i amounts

to successiv<: elimination along the directions .:c 1 , x 2 , ... , :r:P relating to al-

gorithms of the ADM-type. The availability of the cyclic set of parame-

ters gives us hopes to achieve a prescribed accuracy c by 1naking n 0 (c) =

O(ln(l/h) ln(l/c)) iterations, where h = h 1 = h 2 =···=hp is the grid

step. On the other hand, it is no difficult to achieve what we suggest by

means of ATM with the operators

p 1

R1 • v = ~ L., -h 1/" • ·•n ,

cr:::l a

This applies equally well to the proble1n posed above and requires no less

than n 0 (c) iterations, where

n 0 (c) ~ In (2/c) = 0 (-1- ln ~)

3 54 Vh ..;;; c ,

meaning that the asyn1ptotic behavior of ATiVJ becmnes worse in cmnpar-
ison with ADiVI. However, when p = 3 and h > 1/60, that is, the total
nun1ber of the grid nodes < 2.16 · 105 , a smaller number of iterations is
performed during the course of ATM in contrast to ADM with the cyclic
set of para1neters. With regard to the work and storage required, ATM
being rather economical (in 2-2.5 times) offers n1ore advantages than ADM
on any admissible grid no n1atter how it is chosen.

6. ADM for non-self-adjoint operators. The equation we n1ust solve is of
   the fonn

where A 1 and A 2 are non-self-adjoint positive definite operators subject to

the conditions

(34)

l

.11-1 Ci > - -E f::..

Ct

o:=l,2.

As can readily be observed, the second con di ti on is equivalent to the 111-
equality

(35)

Indeed, by 1nerely setting :r = Aay, we are led to