Spectral viscosity approximations to multidimensional scalar conservation laws

Authors:
Gui Qiang Chen, Qiang Du and Eitan Tadmor

Journal:
Math. Comp. **61** (1993), 629-643

MSC:
Primary 35L65; Secondary 65M12

DOI:
https://doi.org/10.1090/S0025-5718-1993-1185240-3

MathSciNet review:
1185240

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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, *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 convergence rate of the usual optimal order one-half.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1993-1185240-3

Keywords:
Multidimensional conservation laws,
spectral viscosity method,
spectral accuracy,
measure-valued solution,
total variation,
convergence,
error estimate

Article copyright:
© Copyright 1993
American Mathematical Society