The Rate of Convergence of Spectral-Viscosity Methods for Periodic Scalar Conservation Laws

A rate of convergence is proven for spectral-viscosity methods for periodic scalar conservation laws. This rate is obtained by showing the discretization error to be small enough that the difference between the solutions of the spectral-viscosity method and the ordinary viscosity method tends to zero in $L^1 $ as the number of discrete modes tends to infinity and the viscosity simultaneously tends to zero at the appropriate rate. The method is also used to obtain an $L^\infty $ bound for the spectral-viscosity approximations to the elasticity equations; convergence, although without a rate, then follows from compensated compactness theory.