Spectral methods for hyperbolic problems

We review the current state of Fourier and Chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations. Throughout the discussion we emphasize recent developments in the area such as spectral penalty methods, the use of filters, the resolution of the Gibbs phenomenon, and issues related to the solution of nonlinear conservations laws such as conservation and convergence. We also include a brief discussion on the formulation of multi-domain methods for hyperbolic problems, and conclude with a few examples of the application of pseudospectral/collocation methods for solving nontrivial systems of conservation laws.