A priori error estimates for nonlinear scalar conservation laws

In this note, we review the recent work of the authors on a priori error estimates for nonlinear scalar conservation laws. A priori error estimates are important because they shed light into the nature of the corresponding numerical scheme. In this note, we show how to use our a priori error estimation technique to study multidimensional monotone schemes defined in non-Cartesian grids. For the so-called consistent schemes, we prove that the L-infinity(0, T; L-1(R-d))-error goes to zero as (Delta x)(1/2) as the discretization parameter Delta x goes to zero; all previous results give a rate of convergence of only (Delta x)(1/4). The loss of 1/4 in the exponent is due to the fact that it is not known how to prove that the total variation of the approximate solution is uniformly bounded. We show that such estimate is not necessary to obtain optimal error estimates. only the structure of the numerical scheme is important.