1549301742-The_Theory_of_Difference_Schemes__Samarskii

612 Econmnical Difference Schein.es for Multidimensional Proble1ns

containing not only the nodes tj = jr of the grid w 7 , but also fictitious

nodes tj+a/pi et= 1,2, ... ,p-l. Let w~ be a set of the nodes of the grid

w~, for which t > 0. Finally, let P(x,t'), where x E wh, t' E w~, be a node

of the (p + 1 )-din1ensional grid 0 = w h x w~; S be the boundary of the grid

0 containing the nodes P(:r, 0) for J: E wh and the nodes P(x, ij+cr/p) for

tj+u/p E w~ and x E lh,o fo1· all Ct= 1,2, ... ,p, j = 0.1. ... ,jo: 0:, be a

set of the nodes P(x, tj+a!J,), where .c E w~.'~ is a near-boundary node in

the direction Xa of the grid Wh.

With these, we proceed to the complete posing of problems related to

fJ and w. The intention is to use the equation for y in the canonical form

(25) in combination with expression (18) for the difference operator Aa not

only at the regular nodes, but also at the irregular nodes:

( 33) [ -T l + -h 1 ( 1 --h* + -h* l )] ); o-.i+o:/p -- h*^1 h Yi c-j+,~/p zo+l ·

a a+ a- a+ CY

+ 1 fj+a/p + ..!:.y-j+(a-l)/p

h* - h a Yi^0 -l T '

a

where Yf:~~{P = y(x(±lo l, lj+a/pl· Fron1 here it seen1s clear that conditions

(29) are satisfied and D(P) = 0. Because of this, Theorem 2 from Chap-

ter 4, Section 2 asserts that for a solution to equation (33) the estinrn.te is

valid:

n1ax [y(P)[ < mpa.i;c [y(P)[.

PE!1+5' ES

Taking into account that

111ax [Y(P)[ = 1nax [[y(x, t')ffr:,

PE!1+5' t' Ew~ _,

where [[y(x)ffc =max [y(x)[,

xEwh

max PES [Y(P)[ = 1nax (max t'Ew' [[μ(x, t')ffc -, , [[u 0 ffc),

T

we eventually get

(34)