The Fourier method for nonsmooth initial data
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- by Andrew Majda, James McDonough and Stanley Osher PDF
- Math. Comp. 32 (1978), 1041-1081 Request permission
Abstract:
Application of the Fourier method to very general linear hyperbolic Cauchy problems having nonsmooth initial data is considered, both theoretically and computationally. In the absence of smoothing, the Fourier method will, in general, be globally inaccurate, and perhaps unstable. Two main results are proven: the first shows that appropriate smoothing techniques applied to the equation gives stability; and the second states that this smoothing combined with a certain smoothing of the initial data leads to infinite order accuracy away from the set of discontinuities of the exact solution modulo a very small easily characterized exceptional set. A particular implementation of the smoothing method is discussed; and the results of its application to several test problems are presented, and compared with solutions obtained without smoothing.References
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M. Y. T. APELKRANS, "Some properties of difference schemes with discontinuities and a new method with almost quadratic convergence," Report #15A, Department of Computer Science, Uppsala University, Uppsala, 1969.
B. FORNBERG, "On high order approximations of hyperbolic partial differential equations by a Fourier method," Rep. No. 39, Department of Computer Science, Uppsala University, Uppsala, 1972.
- Bengt Fornberg, On a Fourier method for the integration of hyperbolic equations, SIAM J. Numer. Anal. 12 (1975), no. 4, 509–528. MR 421096, DOI 10.1137/0712040
- Lars Hörmander, On the existence and the regularity of solutions of linear pseudo-differential equations, Enseign. Math. (2) 17 (1971), 99–163. MR 331124
- Heinz-Otto Kreiss and Joseph Oliger, Comparison of accurate methods for the integration of hyperbolic equations, Tellus 24 (1972), 199–215 (English, with Russian summary). MR 319382, DOI 10.3402/tellusa.v24i3.10634 H.-O. KREISS & J. OLIGER, "Methods for the approximate solution of time dependent problems," GARP Publications Series no. 10, 1973.
- Andrew Majda and Stanley Osher, Propagation of error into regions of smoothness for accurate difference approximations to hyperbolic equations, Comm. Pure Appl. Math. 30 (1977), no. 6, 671–705. MR 471345, DOI 10.1002/cpa.3160300602 G. I. MARCHUK, Numerical Methods in Weather Prediction, Academic Press, New York, 1974.
- Steven A. Orszag, Numerical simulation of incompressible flows within simple boundaries. I. Galerkin (spectral) representations, Studies in Appl. Math. 50 (1971), 293–327. MR 305727, DOI 10.1002/sapm1971504293 S. A. ORSZAG, "Comparison of pseudospectral and spectral approximations," Studies in Appl. Math., v. 51, 1972, pp. 253-269.
- Joel A. Smoller and Michael E. Taylor, Wave front sets and the viscosity method, Bull. Amer. Math. Soc. 79 (1973), 431–436. MR 348282, DOI 10.1090/S0002-9904-1973-13201-X
- Michael Taylor, Pseudo differential operators, Lecture Notes in Mathematics, Vol. 416, Springer-Verlag, Berlin-New York, 1974. MR 0442523
- Michael E. Taylor, Reflection of singularities of solutions to systems of differential equations, Comm. Pure Appl. Math. 28 (1975), no. 4, 457–478. MR 509098, DOI 10.1002/cpa.3160280403 M. TAYLOR, Pseudodifferential Operators, Vol. 2. (To appear.) H.-O. KREISS & J. OLIGER, "Stability of the Fourier method," Stanford Computer Science Report 77-616, 1977. D. GOTTLIEB & E. TURKEL, "On time discretizations for spectral methods," ICASE Report #78-1, 1978.
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Math. Comp. 32 (1978), 1041-1081
- MSC: Primary 65M10
- DOI: https://doi.org/10.1090/S0025-5718-1978-0501995-4
- MathSciNet review: 501995