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Mathematics of Computation

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Total variation and error estimates for spectral viscosity approximations

Author: Eitan Tadmor
Journal: Math. Comp. 60 (1993), 245-256
MSC: Primary 35L65; Secondary 65M06, 65M12, 65M15
MathSciNet review: 1153170
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Abstract: We study the behavior of spectral viscosity approximations to non-linear scalar conservation laws. We show how the spectral viscosity method compromises between the total-variation bounded viscosity approximations-- which are restricted to first-order accuracy--and the spectrally accurate, yet unstable, Fourier method. In particular, we prove that the spectral viscosity method is $ {L^1}$-stable and hence total-variation bounded. Moreover, the spectral viscosity solutions are shown to be $ {\text{Lip}^ + }$-stable, in agreement with Oleinik's E-entropy condition. This essentially nonoscillatory behavior of the spectral viscosity method implies convergence to the exact entropy solution, and we provide convergence rate estimates of both global and local types.

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Keywords: Conservation laws, spectral viscosity method, spectral accuracy, total variation, Lipschitz stability, convergence rate estimates
Article copyright: © Copyright 1993 American Mathematical Society