Convergence of Spectral Methods for Nonlinear Conservation Laws

. We discuss the convergence of Fourier method for scalar nonlinear conservation laws which exhibit spontaneous shock discontinuities. Numerical tests indicate that the convergence may (and in fact in some cases we prove it must) fail, with or without post-processing of the numerical solution. Instead, we introduce here a new kind of spectrally accurate vanishing viscosity to augment the Fourier approximation of such nonlinear conservation laws. Using compensated compactness arguments augmented by the assumption of L 1 -- stability, we show for the inviscid Burgers model equation, that this spectral viscosity method prevents oscillations and convergence to the unique entropy solution follows. Key Words. Burgers equation, entropy solution, the spectral viscosity method, compensated compactness, convergence. AMS (MOS) Subject Classification: 35L65, 65M10, 65M15 1. Introduction. In this paper we study the convergence of spectral methods for nonlinear conservation laws. Specifically...