Super Viscosity And Spectral Approximations Of Nonlinear Conservation Laws

this paper we restrict our attention to periodic problems. For a treatment of the nonperiodic case in terms of the Legendre SV method we refer to [MOT]. The purpose of the SV method is to stabilize the nonlinear spectral approximation without sacrificing its underlying spectral accuracy. This is achieved by augmenting the standard spectral approximation with high frequency regularization. In x3 we briefly review the convergence results of the periodic Fourier SV method, [T2]--[T5], [MT], [CDT], [S]. These convergence results employ high frequency regularization based on second order viscosity. In x4 we introduce spectral approximations based on "super-viscosity", i.e., high-frequency parabolic regularizations of order ? 2. We prove th