1549301742-The_Theory_of_Difference_Schemes__Samarskii

692 Methods for Solving Grid Equations

determination of ktl. In so doing the passage from the kth iteration

to the (k + l)th iteration necessitates the storage not only for the value

t (i 1 h 1 , i 2 h 2 ), but also for the value~ (i 1 h 1 , i 2 h 2 ).

6. A higher-accuracy scheme in a rectangle. In Chapter 4, Section 5, the
   Dirichlet problem

( 45) ~^11 = - f ( x) , J: E^0 , u =^11 ( .'t) , x E r ,

was completely posed, for which the difference schen1e of fourth-order ac-

curacy was composed in the following way:

( 46) A'y=-<p(x), xEwh, y=μ(x), XE/1i,

where

C\'=l,2,

h2 h2
<p --f+- 12 1 A^1. f+- 12 2 A^2.. f
0
Let now H = Q be a space of all grid functions given on the grid w h
and vanishing on the boundary ih. Having involved the operators A y =
-A'y and Ry= -Ay specified for any y EH, we deduce instead of (46)
that A y = <p, where <p f:- <p only at the near-boundary nodes. Applying the
estimates obtained in Chapter 4 for the operator A yields

and reveals the coefficients to be c 1 = ~ and c 2
arranges itself as a sum

1. The operator R

Being concerned with the same quantities b, ~ and w 0 as in Section 5, we

• 0 0 0 0
  might agree to consider / 1 = ~ / 1 and / 2 = I 2 , where / 1, I 2 were specified
  in Section 4:
  0 (j 0 (j
  l 1 = 2 (1 + J1]) ' i2=4v0'