1549301742-The_Theory_of_Difference_Schemes__Samarskii

672 Methods for Solving Grid Equations

\!Vhen n = 2P, where p > 0 is an integer, Gn is found by successively applying

fonnula ( 44) for m = 1, 2' ... '2p-l with the starting point el = { 1}.

Before taking up the general case, we give below the final result of

this procecl ure for n = 16 = 24

()2 = {1,3},

()4 = {1,7,3,5}'

88 ={l,15, 7, 9, 3, 13, 5, 11},

(}16={1,31,15.17,7,25,9.23,3,29, 13,19,5,27, 11,21}.

When moving from Gm to G 2 m, the member G 2 m(2i) = 4m - Gm(i) stands

on the right from the member Gm (i) by the approved rule ( 44).

Among other things, we may attempt an arbitrary positive integer

n > 0 in the form

n = 2 k1 + 2 k2 + ... + 2 k,,

where t > 0 is an integer, ki > are integers such that ki > k;+i + 1,

i = 1, 2, ... , t-l, and k 1 > 0, constitute a new sequence of odd integers

( 47) j=l,2, ... ,t,

and then set n 1 + 1 = 2 n+ l, which formally corresponds to the value k 1 + 1 =

-1. It follows from the foregoing that

nJ·+1 - l = 2 k } -" ~ .1+1 _c) -

--4-- nj.

In terms of how nj+l may be affected, some of the possibilities of interest
are:

The algorithm of obtaining the ordered set Gn is mostly based on the
approved rules for the transition from the sets Gn J to the sets Gn 1+1. Having
formed the set en]' the differences 'i are sought in two ways:

(1) if rj > 2, then the chain of the available sets