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Higher order and spectral methods have been used with success for elliptic and parabolic initial and boundary value problems with smooth solutions. On the other hand, higher order methods have been applied to hyperbolic problems with less success, as higher order approximations of discontinuous solutions suffer from the Gibbs phenomenon. We extend past work and show that spectral methods yield spectral convergence of moments, even when applied to problems with discontinuous solutions. Besides spectral Fourier methods for periodic domains we also prove high order convergence for adjoint-consistent numerical methods, exemplified by the discontinuous Galerkin finite element method.

Accuracy of High Order and Spectral Methods for Hyperbolic Conservation Laws with Discontinuous Solutions