Convergence and accuracy of homogeneous conservative schemes 167
Theorem 1 The accuracy of any scheme (2) with coefficients (16) is of
order 2 in the class of smooth coefficients k(x), q(x), f(x) E cC^2 l[O, 1]:lly-ullc<Mh^2
with constant M > 0 independent of h, whereas it is of order 1 in the class
of discontinuity coefficients k(x), q(x), f(x) E QC^2 l[O, 1]:lly-ullc<Mh.
The best scheme also has the second-order accuracy in the class of discon-
tinuity coefficients.
These assertions follow from the representation of the approximation
error in the form (6)-(8) and a priori estimate (12). On the basis of the
estimates for 7); and 1/;{ obtained in Section 3.2 we find that
(16)yielding(17)Combination of relations (16)-(17) just established and estimate (12) pro-
vides the sufficient background for the validity of the assertions of the the-
orem in light of the asymptotic representations
1/;~ + 1/;~+1 = 0(1),
0 0(^01) · + (^01) · = O(h) for any e E [O 1]
'+'n '+'n+l ' '
7/n+l = 0(1),
In the case of smooth coefficients we achieve for any admissible scheme
7); = O(h^2 ) and 1/;{ = O(h^2 ) for all i = 1, 2, ... , N - 1 and, therefore,
(1, I 77ll + (1, Iμ ll = O(h^2 ).
Remark It can be shown that the approximation in the class of smooth co-
efficients is necessary and sufficient for the convergence of the homogeneous
scheme (2).