348 Difference Schen1es with Constant CoefficientsWe are going to show that the scheme(15)2 h2 + h2
Yt=LAa(O"aY+(l-O"a)Y)+^1
12(^2) AiA2y+cp,
a=l
y(x, 0) = u 0 (x),
with the ingredients
( 16)
O" a =
1
2 l2T
et=l,2,
provides an approximation of 0( I h 14 + r^2 ) in light of the representation of
the residual
This can be done by inserting the well-established expansions
et=l,2.
The outcorne of this is
with O"°' still subject to the first condition (16). In order to prove the
convergence of this scheme with the rate 0(1h1^4 + r^2 ), it is necessary to
obtain an a priori estimate for a solution of the proble1n with zero initial
and bonndary conditions. Having no opportunity to touch upon this topic