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 partial differential equation with a discontinuous solution, Fourier spectral method produces poor pointwise 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 find that the moments with respect to analytic functions are no longer very accurate. However the numerical solution does contain accurate information which can be extracted by a post-processing based on Gegenbauer polynomials. 1Research supported by ARO grant DAAL03-91-G-0123 and DAAH04-94-G-0205, NSF grant DMS-9211820, NASA grant NAGl-lI45 and contract NAS1-19480 while the first author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research (;enter, Hampton, VA 23681-0001, and AFOSR grant 93-0090.