Spectral methods for periodic initial value problems with nonsmooth data
HTML articles powered by AMS MathViewer
- by Pravir Dutt and A. K. Singh PDF
- Math. Comp. 61 (1993), 645-658 Request permission
Abstract:
In this paper we consider hyperbolic initial value problems subject to periodic boundary conditions with nonsmooth data. We show that if we filter the data and solve the problem by the Galerkin-Collocation method, recently proposed by us, then we can recover pointwise values with spectral accuracy, provided that the actual solution is piecewise smooth. For this we have to perform a local smoothing of the computed solution.References
- S. Abarbanel, D. Gottlieb, and E. Tadmor, Spectral methods for discontinuous problems, Numerical methods for fluid dynamics, II (Reading, 1985) Inst. Math. Appl. Conf. Ser. New Ser., vol. 7, Oxford Univ. Press, New York, 1986, pp. 129–153. MR 875458
- Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, and Thomas A. Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988. MR 917480, DOI 10.1007/978-3-642-84108-8
- P. Dutt, Spectral methods for initial-boundary value problems—an alternative approach, SIAM J. Numer. Anal. 27 (1990), no. 4, 885–903. MR 1051112, DOI 10.1137/0727051
- P. Dutt and A. K. Singh, The Galerkin-collocation method for hyperbolic initial-boundary value problems, J. Comput. Phys. 112 (1994), no. 2, 211–225. MR 1277274, DOI 10.1006/jcph.1994.1093
- David Gottlieb and Eitan Tadmor, Recovering pointwise values of discontinuous data within spectral accuracy, Progress and supercomputing in computational fluid dynamics (Jerusalem, 1984) Progr. Sci. Comput., vol. 6, Birkhäuser Boston, Boston, MA, 1985, pp. 357–375. MR 935160
- Andrew Majda, James McDonough, and Stanley Osher, The Fourier method for nonsmooth initial data, Math. Comp. 32 (1978), no. 144, 1041–1081. MR 501995, DOI 10.1090/S0025-5718-1978-0501995-4 B. Mercier, Analyse numérique des méthodes spectrales, Note CEA-N-2278 (Commissariat a l’Energie Atomique Centre d’Etudes de Limeil, 94190 Villeneuve-Saint Georges).
- Jeffrey Rauch, ${\cal L}_{2}$ is a continuable initial condition for Kreiss’ mixed problems, Comm. Pure Appl. Math. 25 (1972), 265–285. MR 298232, DOI 10.1002/cpa.3160250305
- Michael E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, No. 34, Princeton University Press, Princeton, N.J., 1981. MR 618463
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Math. Comp. 61 (1993), 645-658
- MSC: Primary 65M70
- DOI: https://doi.org/10.1090/S0025-5718-1993-1195431-3
- MathSciNet review: 1195431