1549301742-The_Theory_of_Difference_Schemes__Samarskii

494 Homogeneous Difference Schemes for Tirne-Dependent Equations

and a time grid w 7 = {tj = jr, j = 0, 1, ... } on the segment 0 < t < T to

expound exploratory devices for obtaining difference schemes by analogy

with Chapter 3.

The operator L is approxin1ated by the difference operator

A(f) U; = ~ xi (x;_ 1 ; 2 a; ux,i)x. ,z ~Lu, where a;= a(x;, [),

thereby establishing a correspondence between equation (83) and one of the

weighted schemes

cf tj = A(i) 1'/(cr) • + 'r· •r 'P = f ( ,i; , t).

The cornplete posing of this includes the difference boundary condition

at the point x = 0. The methodology of Chapter 3 furnishes the justifica-

tion of the forthcoming substitutions into the stationary equation: we first

replace Yr by y~^0 ) and f , (O) by ( f - ~ ou) and then ou ~ by u 1 and u by

ui x=O ui

y. The outcome of this is

·) (a) _ h h (a)

a 1 (i Yx,O - 4 Yt,o - 4 fo ,

which admits an alternative form of writing

_ 4 I\ (a)

Yt,o - h al (tJ Yx,o +'Po,

In this regard, we observe that it is po.5si ble to insert f ( 0, i) in place of JJ^0 ).
vVhen the conditionc; for .r = 1 and t = 0 are put together with the
preceding equation, their collection constitutes what is called the difference
boundary-value proble111

y 1 =A(_i)y(^0 l+'f!, O<x=ih<l, t=jr>O,

where

• 4 -
  A(t)y = h a 1 (i)Yx for x=O ,

A(f)y=;~(xa(x,i)y,r)r for O<x=ih<l, x=x;-0.5h.