Spectral Methods for Discontinuous Problems

Spectral methods have emerged as powerful computational techniques for simulation of complex, smooth physical phenomena. Among other applications they have contributed to our understanding of turbulence by successfully simulating incompressible turbulent flows, have been extensively used in meteorology and geophysics, and have been recently applied to time domain electromagnetics. Several issues arise when applying spectral methods to problems which feature sharp gradients and discontinuities. In the presence of such phenomena the accuracy of high order methods deteriorates. This is due to the well known Gibbs phenomenon that states that the pointwise convergence of global approximations of discontinuous functions is at most first order. In the presence of a shock wave global approximations are oscillatory and converge nonuniformly. Recent advances in the theory and application of spectral methods indicate that high order information is retained in stable spectral simulations of discontinuous phenomena and can be recovered by suitable postprocessing techniques.