1549301742-The_Theory_of_Difference_Schemes__Samarskii

608 Eco1101nical Difference Schernes for Multidimensional Problems

et= 1, 2, ... , p. In determining yj+cx/p we inm;t solve the boundary-value

proble1n

A· ZaY-icx-1 "lJ+cx/p - C, ZaY 1J+c,/Jl +A lo. +i"lJ+cx/p Yio.+l = -F ex~!-1.i+a/p for x E wh,,

(24)

for x E --v I h,cx > CY=l,2, ... ,]J,

with varying subscripts only. The difference equation is written on the

segment 6a E Ccx with the endpoints belonging to the boundary ih , c, and

it can be solved by the elimination method along all the segments 6cx for

fixed Cl'. For doing so our expenses are not considerable, since the number

of arithmetic operations required at every node of the grid wh are 0(1)

solely in connection with successive determination of the values yj +l/p,

yJ+^2 f1', ... ,yj+cxfp, ... ,yj+l by setting o: = 1,2, ... ,p and changing the

directions of the eli111inations. Thus, the locally one-dimensional sche111e

(21 )-(23) falls within the category of econ01nical schemes.

6. The error of approxin1ation of a locally one-dimensional schen1e. Upon
   raising the question regarding the error of approximation provided by one
   or another LOS it is straightforward to verify that every separate equation
   (21) with the number Cl' does not approxi1nate equation (15) in spite of the
   fact that the sum of the residuals 1/• = 1/; 1 + 1/• 2 + · · · + 1/;r involved tends to
   U as T -· 0 and !hi - 0.
   Let -u = u( x, t) be a solution of proble111 ( 15) with the operator Lau, =
   82 u/ox; and yJ+cx/p, Cl' = l, 2, ... , p be a solution of proble1n (21)-(23).
   The accuracy of LOS is characterized, as usual, by the difference 11+^1
   uJ+l = zi+l.
   The intermediate values yj+cx/p need to be compared with uj+cx/p
   u(x, tj+ajp) by making the choice zj+cx/p = yJ+cx/p - uj+cx/p. Upon sub-

stituting yi+cx/p = zj+a/p + uj+cx/p into equation (21) we may set up the
problen1 for the error zi +^1 :

(25)

where

(26)

zj+cx/p _ zj+(cx-l)/p

_________ =A zj+a/p + .. t.J+cx/p

T o lf/ ex '

j=O,l, ... ,J 0 , Cl'=l,2, ... ,p,

Z.j+ufr = U f'o1· ;• r E lh,o > z(.1:. U) = 0.

uj+o/p - uj+(a-l)/p

iJ.,j+a/p. = Acx uj+cx/p + .,J+cx/p ~ex - ---------

T