1549301742-The_Theory_of_Difference_Schemes__Samarskii

648 Methods for Solving Grid Equations

or, what amounts to the same,

2v(ll - h 2 2 v~.r1:r1^1 ) - r 'ti vCll = v(l-l) ' h 1 <x 1 <1 1 - h 1 ,

The preceding typical equation can be solved by the elimination lnethocl

with regard to three-point equations. In giving FY) the intention is to use

the formula

( 10) yCkJ J = c(kJ PCkJ J + qCkJ J

with new vectors p)k) and qy) still subjecL to (8):

( 11) cc k) PJ ( k) + ( qi k) -- cc k - 1) [ ( P 1 k _- 2 k-1 1) + ( Pi+k - 2 1) k-1 + ( qi k - J) l

+ (c

(k-1))2 P(k-1) + (k-1) + (k-1)

1 q 1- 2 k-1 q.+J 2 k-1.

Granting the decomposition

(12)

and taking into acount that

cCkJ = (c<k-1))2 - '2 E,

we recast equation (11) as

(c(k-1J)2 ( P 1 CkJ P 1 Ck-lJ) = cCk-lJ ( P 1 Ck-1J 2 k-1 + Pi+Ck-lJ 2 k-1 + q 1 Ck-1J)

with further elin1ination of cCk-^1 l. The outco1ne of this is

CCk-1J^51 ck-1J = P 1 Ck-11 2 k-1 + Pi+Ck-1J 2 k-1 + q 1 (k-1J , Pj uJ - Pj rk-1) + sCk-11 j

with q)k) still subject to (12). Substitution of forrnula (10) into (7) yields

C (k-1) [ Yi - Pj (k-1)] =qi (k-1) + Yj_

2 k-1 + Yi+ 2 k-1,