A priori error estimates for numerical methods for scalar conservation laws part III: Multidimensional flux-splitting monotone schemes on non-cartesian grids

This paper is the third of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed. In this paper, we consider multidimensional flux-splitting monotone schemes denned on non-Cartesian grids. We identify those schemes which are consistent and prove that the L^∞ (0, T; L¹ (ℝ^d))-norm of the error goes to zero as (Δx)^1/2 when the discretization parameter δx goes to zero. Moreover, we show that nonconsistent schemes can converge at optimal rates of (Δx)^1/2 because (i) the conservation form of the schemes and (ii) the so-called consistency of the numerical fluxes allow the regularity properties of the approximate solution to compensate for their lack of consistency.