A priori error estimates for numerical methods for scalar conservation laws .1. The general approach

In this paper, we construct a general theory of a priori error estimates for scalar conservation laws by suitably modifying the original Kuznetsov approximation theory. As a first application of this general technique, we show that error estimates for conservation laws can be obtained without having to use explicitly any regularity properties of the approximate solution. Thus, we obtain optimal error estimates for tile Engquist-Osher scheme without using the fact (i) that tile solution is uniformly bounded, (ii) that the scheme is total variation diminishing, and (iii) that tile discrete semigroup associated with the scheme has the L(1)-contraction property, which guarantees an upper bound for tile modulus of continuity in time of the approximate solution.