Convergence of Finite Difference Schemes for Conservation Laws in Several Space Dimensions : the Corrected Antidiffusive Flux Approach

In this paper, we apply the general method we have presented elsewhere and prove the convergence of a class of explicit and high-order accurate finite difference schemes for scalar nonlinear hyperbolic conservation laws in several space dimensions. We consider schemes constructed—from an £-scheme— by the corrected antidiffusive flux approach. We derive "sharp" entropy inequalities satisfied by both ^-schemes and the high-order accurate schemes under consideration. These inequalities yield uniform estimates of the discrete space derivatives of the approximate solutions, which are weaker than the so-called BV (i.e., bounded variation) estimates but sufficient to apply our previous theory.