Abstract
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.
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References
Abarbanel, A., Gottlieb, D., and Tadmor, E. (1986). Spectral methods for discontinuous problems, in Morton, W., and Baines, M. J. (eds.),Numerical Methods for Fluid Dynamics II, Oxford University Press, London, pp. 129–153.
Cai, W., Gottlieb, D., and Shu, C.-W. (1992). On One-Sided Filters for Spectral Fourier Approximations of Discontinuous Functions,SIAM J. Numer. Anal. 29, 905–916.
Canuto, C., Hussaini, M. Y., Quarternoni, A., and Zang, T. A. (1988).Spectral Methods in Fluid Dynamics, Springer-Verlag.
Gottlieb, D., and Orszag, S. (1977).Numerical Analysis of Spectral Methods: Theory and Applications, SIAM-CBMS, Philadelphia.
Gottlieb, D., and Tadmor, E. (1985). Recovering Pointwise Values of Discontinuous Data Within Spectral Accuracy, in Murman, E. M., and Abarbanel, S. S. (eds.),Progress and Supercomputing in Computational Fluid Dynamics, Birkhäuser, Boston, pp. 357–375.
Gottlieb, D., and Shu, C.-W. (1994). Resolution properties of the Fourier method for discontinuous waves,Meth. Appl. Mech. Engin. 116, 27–37.
Gottlieb, D., and Shu, C.-W. (1993). On the Gibbs Phenomenon III: Recovering exponential accuracy in a sub-interval from the spectral partial sum of a piecewise analytic function, ICASE Report No. 93-82, NASA Langley Research Center,SIAM J. Numer. Anal. (to appear).
Gottlieb, D., and Shu, C.-W. (1995). On the Gibbs Phenomenon IV: Recovering exponential accuracy in a sub-interval from the Gegenbauer partial sum of a piecewise analytic function,Math. Comp. 64, 1081–1095.
Gottlieb, D., and Shu, C.-W. (1994a). On the Gibbs Phenomenon V: Recovering exponential accuracy from collocation point values of a piecewise analytic function, ICASE Report 94-61, NASA Langley Research Center,Numer. Math., to appear.
Gottlieb, D., Shu, C.-W., Solomonoff, A., and Vandeven, H. (1992). On the Gibbs Phenomenon I: recovering exponential accuracy from the Fourier partial sum of a non-periodic analytic function,J. Comput. Appl. Math. 43, 81–92.
Kreiss, H., and Oliger, J. (1979). Stability of the Fourier Method,SIAM J. Numer. Anal. 16, 421–433.
Maday, Y., and Tadmor, E. (1989). Analysis of the spectral vanishing viscosity method for periodic conservation laws,SIAM J. Numer. Anal. 26. 854–870.
Maday, Y., Ould Kaber, S., and Tadmor, E. (1993). Legendre pseudospectral viscosity method for nonlinear conservation laws,SIAM J. Numer. Anal. 30, 321–342.
Madja, A., McDonough, J., and Osher, S. (1978). The Fourier Method for Nonsmooth Initial Data,Math. Comput. 32, 1041–1081.
Tadmor, E. (1989). Convergence of spectral methods for nonlinear conservation laws,SIAM J. Numer. Anal. 26, 30–44.
Vandeven, H. (1991). Family of Spectral Filters for Discontinuous Problems,J. Sci. Comput. 8, 159–192.
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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.
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Shu, CW., Wong, P.S. A note on the accuracy of spectral method applied to nonlinear conservation laws. J Sci Comput 10, 357–369 (1995). https://doi.org/10.1007/BF02091780
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DOI: https://doi.org/10.1007/BF02091780