Running Head. Multidimensional Spectral Viscosity Approximations

We study the spectral viscosity (SV) method in the context of multidimensional scalar conservation laws with periodic boundary conditions. We show that the spectral viscosity, which is suÆciently small to retain the formal spectral accuracy of the under-lying Fourier approximation, is large enough to enforce the correct amount of entropy dissipation (which is otherwise missing in the standard Fourier method). Moreover, we prove that because of the presence of the spectral viscosity, the truncation error in this case becomes spectrally small, independent of whether the underlying solution is smooth or not. Consequently, the SV approximation remains uniformly bounded and converges to a measure-valued solution satisfying the entropy condition, that is, the unique entropy solution. We also show that the SV solution has a bounded total vari-ation, provided that the total variation of the initial data is bounded, thus conrming its strong convergence to the entropy solution. We obtain an L 1 convergence rate of the usual optimal order one-half. Key words. Multidimensional conservation laws, spectral viscosity method, spectral accuracy, measure-valued solution, total variation, convergence, error estimate AMS(MOS) subject classication. 35L65, 65M10, 65M1