Analysis of the Spectral Vanishing Viscosity Method for Periodic Conservation Laws

The convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid Burgers’ equation is analyzed. It is proved that this kind of vanishing viscosity is responsible for spectral decay of those Fourier coefficients located toward the end of the computed spectrum; consequently, the discretization error is shown to be spectrally small, independently of whether or not the underlying solution is smooth. This in turn implies that the numerical solution remains uniformly bounded and convergence follows by compensated compactness arguments.