1549301742-The_Theory_of_Difference_Schemes__Samarskii

Operator-difference schemes 387

4. The canonical form of three-layer schemes. We may attempt the three-
   layer scheme (2) in the canonical form

(6) B Yn+l 2 - T Yn-1 + R ( Yn+l -^2 Yn + Yn-1 ) + A Yn = 'P ·

By comparing (6) with (2) we see that such a writing is always possible if

we agree to consider

Also, it will be sensible to introduce the notations

Yt - Yr

Yrt =

y-y

yo =

t 2 T T

and regard the equation

(8)

Byo t + T^2 RYrt +Ay = <p(t),

0 < t = n TE W 7 , y(O) = Yo ,

[;-2y+y

T2

Y( r) = Y1 ,

to the canonical form of a three-layer scheme together with (6).

Example 2 We now turn to the weighted three-layer scheme

(9)

and try to reduce it to the canonical form. With this aim, we make use of

the formulae

y-y y-2y+y T2

y=y+ 2 + 2 Y + T Yj + 2 Yt t ,

Y - Y Y - 2 y + y r^2

y - 2 + 2 = y - T Yj + 2 Yr t ,

Upon substituting these expressions into (9) we write the weighted scheme
in the canonical form (8), where

(10) R