The Fourier method for nonsmooth initial data
Authors:
Andrew Majda, James McDonough and Stanley Osher
Journal:
Math. Comp. 32 (1978), 10411081
MSC:
Primary 65M10
MathSciNet review:
501995
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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.
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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.
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 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.
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 L. HÖRMSNDER, "On the existence and the regularity of solutions of linear pseudodifferential equations," Enseignement Math., v. 17, 1971, pp. 99163. MR 0331124 (48:9458)
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 H.O. KREISS & J. OLIGER, "Comparison of accurate methods for the integration of hyperbolic equations," Tellus, v. 24, 1972, pp. 199215. MR 0319382 (47:7926)
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 H.O. KREISS & J. OLIGER, "Methods for the approximate solution of time dependent problems," GARP Publications Series no. 10, 1973.
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 A. MAJDA & S. OSHER, "Propagation of error into regions of smoothness for accurate difference approximations to hyperbolic equations," Comm. Pure Appl. Math., v. 30, 1977, pp. 671705. MR 0471345 (57:11080)
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 G. I. MARCHUK, Numerical Methods in Weather Prediction, Academic Press, New York, 1974.
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 S. A. ORSZAG, "Numerical simulation of incompressible flows within simple boundaries, I: Galerkin (spectral) representations," Studies in Appl. Math., v. 50, 1971, pp. 293327. MR 0305727 (46:4857)
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 S. A. ORSZAG, "Comparison of pseudospectral and spectral approximations," Studies in Appl. Math., v. 51, 1972, pp. 253269.
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 J. SMOLLER & M. TAYLOR, "Wave front sets and the viscosity method," Bull. Amer. Math. Soc., v. 79, 1973, pp. 431436. MR 0348282 (50:780)
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 M. TAYLOR, Pseudodifferential Operators, Lecture Notes in Math., vol. 410, SpringerVerlag, Berlin, 1974. MR 0442523 (56:905)
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 M. TAYLOR, "Reflection of singularities of solutions to systems of differential equations," Comm. Pure. Appl. Math., v. 28, 1975, pp. 457478. MR 0509098 (58:22994)
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 M. TAYLOR, Pseudodifferential Operators, Vol. 2. (To appear.)
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 H.O. KREISS & J. OLIGER, "Stability of the Fourier method," Stanford Computer Science Report 77616, 1977.
 [16]
 D. GOTTLIEB & E. TURKEL, "On time discretizations for spectral methods," ICASE Report #781, 1978.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197805019954
PII:
S 00255718(1978)05019954
Keywords:
Fourier method,
hyperbolic equations,
Cauchy problem,
smoothing techniques,
convergence rate,
stability
Article copyright:
© Copyright 1978
American Mathematical Society
