Convergence of Spectral Methods for Nonlinear Conservation Laws

Abstract

We discuss the convergence of Fourier methods for scalar nonlinear conservation laws that 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^\infty $-stability, we show for the inviscid Burgers’ model equation that this spectral viscosity method prevents oscillations, and convergence to the unique entropy solution follows.