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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.

Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach