Total Variation and Error Estimates for Spectral Viscosity Approximations

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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.