1549301742-The_Theory_of_Difference_Schemes__Samarskii

482 Homogeneous Difference Sche1nes for Thne-Dependent Equations

vve arrive at

xn+I/2

)

J

o'll 11 _ _ _ a

(59 Dt(x, t! dx = w(xn+I/ 2 , t) - w(xn-I/2> t) - Q(t) + h <p n.

·"n-1/2

The usual transition to a. difference equation lea.els to

Yt = Ay(a) + <p + Q~[) for x = Xn,

where Ay = (ay_ 75 )x·

Sun11na.rizing, the difference scheme for the problen1 described by (1 )-

(3), (58) is of the forn1

(60) Yt -- Ay (a) + <p + h^1 Q(t) - oi,n, 0 < x = X; < 1, t = tj > 0,

Y 0 =μ 1 , YN=μ 2 , y(x,0)=u 0 (x),

where D; n is, as usual, Kronecker's delta..

For ' the error z = y - u we have equation with the right-hand side

(61)

and the boundary conditions

( 62) z(x,0)=0.

Ou account of the balance equation (59) the residual is representable by

(63) 'ljJ = 1)x + ijJ* '

(64)

Here we adopt v; = v(x;_ 0 G> t) as was clone before. For the sake of sim-
plicity we take () < 0.5, that is, .1:n < ~ < Xn+l/ 2 , pern1ittiug us to deduce
that

1);=0(h^2 +rm~) fora.11 if:.n+l,

4,7 = O(h^2 + r^2 ) for all i # n.