1549301742-The_Theory_of_Difference_Schemes__Samarskii

The summarized approximation method 633

if p = 2 or

0

( E - :~ A2) (y2/3 + lta) = 2 yi/3 + F'2'

Fi = u 0 + T 3 u_ 0 + T^2 Ai U 0 + T 2 ( 3 2. f 1 - 6 1 ( All+ f ') ) t=O,

F2 = T^2 (~ f2 - ~ (A 1l + !) ) if p = 3.

3 9 t =ll

We are now interested in the more detailed designs of LOS for p = 2:

2

(75) ll(x, 0) = 1l 0 (x), ( E - r

4

Ai) yi/^2 =Fi for t = 0.5 T,

(76)

. Yj+i_2y1+i12+Yj - l A (.,J+t. .J).^1 J
T 2 - 4 :! !J + y + 2 <p 2 1 j = 1, 2, ....

In that. case the boundary conditions becorne

y^1 +i/2 - μ( x t ) for ' i

• J J+l/'.J x E lh'
  (77). i
  y^1 + = μ(x J t 1+i ) for x E /h'^2

and the function y1+^1!^2 can be recovered from the equation

with the right-hand side <P{ and the known boundary conditions (77). In

turn, the function yi +i is found from the equation

2

(^0) y j+J - -T A 2. y j+J - "'2 ;.}+i/2
4
with the right-hand side <P~+l/^2 and the known boundary conditions (77).
Either of these equations can be solved by the standard elimination 1nethod.