A note on the accuracy of spectral method applied to nonlinear conservation laws

Fourier spectral method can achieve exponential accuracy both on the approximation level and for solving partial differential equations if the solutions are analytic. For a linear PDE with discontinuous solutions, Fourier spectral method will produce poor point-wise accuracy without post-processing, but still maintains exponential accuracy for all moments against analytic functions. In this note we assess the accuracy of Fourier spectral method applied to nonlinear conservation laws through a numerical case study. We have found out that the moments against analytic functions are no longer very accurate. However the numerical solution does contain accurate information which can be extracted by a Gegenbauer polynomial based post-processing.Research supported by ARO Grant DAAL03-91-G-0123 and DAAH04-94-G-0205, NSF Grant DMS-9211820, NASA Grant NAG1-1145 and contract NAS1-19480 while the first author was in residence at ICASE, NASA Langley Research Center, Hampton, Virginia 23681-0001, and AFOSR Grant 93-0090.