ABSTRACT
We study the notions of consistency and accuracy of a large family of methods (Finite Volume Methods) for numerically solving conservation laws. Two types of difficulties arise: first, the solutions have low regularity (shock waves); second, when working on nonuniform non-Cartesian grids, consistency is lost. It is shown that even when consistency is lost due to irregularities of the grid, not only are the methods still convergent, but no loss in the optimal convergence rate occurs (supraconvergence). In other words, the optimal rate, here O(Δ1/2), is preserved provided (i) the algorithm is in conservation form, (ii) the numerical flux is consistent (not to be confused with the consistency of the scheme). This is one of the first supraconvergence results for low regularity (BV) problems.
