1549301742-The_Theory_of_Difference_Schemes__Samarskii

500 Homogeneous Difference Schernes for Time-Dependent Equations

for fixed t E wT through the approxirnation Av + 'P = (a.( x, t) u 5 J .f: + 'P to

Lv + f. The forthcoming replacements 82 v/ot^2 lt=tj ~ vrt and Lv + f ~

A(tj) 1/"^1 .a~)+ tp, where

v(a,,a2) -- (J 1 ·u + (1-(J 1 - (J) 2 v + (J 2 -ll '

ll = llj , 1l = llj-l,

complement subsequent constructions. As a final result we obtain the ho-

mogeneous weighted three-layer scheme

(5)

The coefficient a. can be taken on the middle layer t = tj.

Substituting y = y + ryt + 0.5T^2 Yrt and y = y - ryt + 0.5T^2 Yrt, where

yo t = (y - y)/(2r) and Ytt = (y - 2y + y)/r^2 , into (5) yields

which adrni ts an alternative form

where Eis the identity operator. For (J 1 = (J 2 = (J it becomes the syn1metric
scheme

(7) (E-(Jr^2 A)yr 1 =Ay+tp(x,t), O<i=jr,

which will be given special investigation in the sequel.
Before going (urther, we must append to (7) the initial and boundary
conditions. These depend on the range of variables; thus, the boundary
conditions and the first initial condition are specified exactly:

( 8) y( 0, t) = 1l 1 ( t) , y( 1, t) = 1l 2 ( t) , y( x, 0) = v 0 ( x).

There are two ways of approximating the second initial condition
01l I ot lt=:O = Uo ( x), one of which generates an approximation of order 2
m r: