1549301742-The_Theory_of_Difference_Schemes__Samarskii

The summarized approximation 1nethod

where

t' = tj'+a/p I

llYll C =max xEwh IYI,

= 1nax

:-i:Ew* h

llYll C-y =max XE"fh IYI,

=max 0

:cEw h

611

To prove this assertion, we represent a solution of the proble1n con-
cerned as a sum
y=y+v+w,
where y is a solution to homogeneous equations (21) with boundary and
initial conditions (22)-(23) ctnd v and ware solutions to nonhomogeneous
equations (21) with the hon1ogeneous boundary and initial conditions:

vj+a/p - vJ+(a-l)/p. o.

---------=A a v;+a/p + cp1+a/p Ct' '

T

( 31)

v(x,0)=0, vj+a/p =^0 for x E -v I h Ci I

'

CY=l,2, ... ,p,

(32)

wj+a/p - wj+(a-l)/p ' '

----------=Ao: wl+<x/p + cp:J+u/p, XE wh,

T

cx=l,2, ... ,p, w(x,0)=0, Wj+a/p =^0 for· x E"' I h ' (Y '

0

Here cp °' and cp: are specified by the formulas

'Pu = {

0

cp: = {

0

'Pa for :i: E w 1 ,,^0 for x E w h I

0 for x E w* h' cp Ci for x E w· h'

so that 0

'Pa+cp:=cpa for xEwh,

thereby clarifying that cp: differs frmn zero only at the near-boundary
nodes.
For convenience in analysis, the grid w~ is made up by

w~ = {O I tj+a/p = (j + cx/p) TI j = 0, 1, 2, ... ,jo - 1, CY= 1, 2, ... ,p} I