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

DOI:
https://doi.org/10.1090/S0025-5718-1993-1153170-9

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 -stable and hence total-variation bounded. Moreover, the spectral viscosity solutions are shown to be -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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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1993-1153170-9

Keywords:
Conservation laws,
spectral viscosity method,
spectral accuracy,
total variation,
Lipschitz stability,
convergence rate estimates

Article copyright:
© Copyright 1993
American Mathematical Society