1549301742-The_Theory_of_Difference_Schemes__Samarskii

Other problems 185

As can readily be observed from the foregoing, Ai > 0, B; > 0 and D; ~ 0,

since bi < 0, bt > 0 and d; > 0.

Equations (22) can be solved by the elimination method for any hand

r. The error of approximation for scheme (21)

is representable by ij; = 1fJ(l) + ·ij;(^2 ) with the members

'!/;(!) = [(aui') x - du+ <p] - [(ku^1 )^1 - qu + f],

The first summand satisfies the estimate

for k E C(^3 l, q, f E C(^2 l.

By virtue of the relations

we obtain

aui' =kit' - ~ h(ku^1 )^1 + O(h^2 ),

a(+l)u"' = ku' + ~ h(ku')' + O(h^2 ),

(aux:),,.= (ku')' + O(h^2 )

'i/;(^2 ) = - R (ku')' + R(ku')' + O(h^2 ) = R

2

(ku')' + O(h^2 ) = O(h^2 )

R+l R+l

exploiting the fact that R = 0.5 h Ir l/k = O(h).
Because of this, the monotone scheme (21) generates an approximation
of order 2:

(23)