Abstract: A finite difference scheme for a two-dimensional scalar conservation law cannot have conservation form, a fixed and finite stencil, formal second order accuracy, and be total variation decreasing (TVD) - this is a well known theorem of Goodman and Leveque. This paper investigates the possibility of relaxing the restriction on the stencil in order to construct a scheme that is conservative, has second order accuracy, and is TVD. A non local limiter is introduced, which operates on either flux corrections (for flux limiter schemes) or slopes (for slope limiter schemes) in order to enforce the TVD property. The main results of the paper are compatibility of the nonlocal limiter with formal second order accuracy and convergence of the resulting approximations (modulo extraction of a subsequence) to a weak solution of the conservation law.
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