Viscosity Approximations to Multidimensional Scalar Conservation Laws

Authors: Gui-Qiang Chen, Qiang Du, and Eitan Tadmor

Title: Viscosity Approximations to Multidimensional Scalar Conservation Laws

Abstract
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 sufficiently small to retain the formal spectral accuracy of the underlying 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, {\it 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 variation, provided that the total variation of the initial data is bounded, thus confirming its strong convergence to the entropy solution. We obtain an $L^{1}$ convergence rate of the usual optimal order one-half.
This article has appeared in:
Mathematics of Computation, vol. 6, pages 629-643 (1993)
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Author Address
    Gui-Qiang Chen
    Department of Mathematics
    Northwestern University
    Evanston, IL 60208-2730
    gqchen@math.nwu.edu