1549301742-The_Theory_of_Difference_Schemes__Samarskii

690 Methods for Solving Grid Equations

with the supplementary conditions

k+l k k k k

y = y = 0 on /h, v = y for x E wh, v = p. for x E lh.

A solution kt! of problem (38) on wh +ih can be obtained through the use

of such a procedure: k+l v = k+l y on wh and k+l v = p. on lh. The recovery of

k+i y reqmres. successive. so 1. utron o t e f h pro bl ems

( 39)

k

(E + w 0 Ri) Y = F , (E, + w 0 R2 ) k+l Y = Y,

where

( 40) R 1 y = Y - h2 Yi1-1 + Y - h2 Yi^2 -1

1 2

( 41) R 2 y __ - Yi1+1 h2 - Y _ Yi2+1 h2 - Y

1 2

In preparation for this, we have at our disposal

The operator E+w 0 R 1 is specified on a three-point pattern ( i 1 h 1 , i)i 2 ),

((i 1 - 1) h 1 , i 2 h 2 ), (i 1 h 1 , ((i 2 - 1) h 2 ), while the operator E + w 0 R2 - on

a three-point pattern (i 1 h 1 , i 2 h 2 ), ((i 1 + 1) h 1 , i 2 h 2 ), (i 1 h 1 , ((i 2 + 1) h 2 ).
Upon substituting the above expressions for R 1 y and R 2 y into (39) we
establish the recurrence relations which will be needed in the sequel:

( 42) y=

( 43)

k

X1 Yi1-l + X2 Yi2-l + F

l + x 1 + X 2

k+l k+l

X1 Yi1+1+x2 Yi 2 +1+Y

1 + x 1 + X 2

fJl,,h = 0,

k+I

y l,,h = 0,