On a high order numerical method for functions with singularities
Author:
Knut S. Eckhoff
Journal:
Math. Comp. 67 (1998), 10631087
MSC (1991):
Primary 65M70, 65N35
MathSciNet review:
1459387
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Abstract: By splitting a given singular function into a relatively smooth part and a specially structured singular part, it is shown how the traditional Fourier method can be modified to give numerical methods of high order for calculating derivatives and integrals. Singular functions with various types of singularities of importance in applications are considered. Relations between the discrete and the continuous Fourier series for the singular functions are established. Of particular interest are piecewise smooth functions, for which various important applications are indicated, and for which numerous numerical results are presented.
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 K. P. Bube, convergence of trifonometric interpolants. SIAM J. Numer. Anal. 15, (1978), pp. 12581268. MR 80g:42002
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 P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, FL, (1984). MR 86d:65004
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 K. S. Eckhoff, Accurate and efficient reconstruction of discontinuous functions from truncated series expansions. Math. Comp. 61, (1993), pp. 745763. MR 94a:65073
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 K. S. Eckhoff, On discontinuous solutions of hyperbolic equations. Comput. Methods Appl. Mech. Engrg. 116, (1994), pp.103112. MR 95c:65163
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 K. S. Eckhoff, Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions. Math. Comp. 64, (1995), pp. 671690. MR 95f:65234
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 K. S. Eckhoff, On a high order numerical method for solving partial differential equations in complex geometries. J. Scient. Comp. 12 (1997), pp. 119138.
 12.
 K. S. Eckhoff and J. H. Rolfsnes, A Fourier method for nonsmooth hyperbolic problems. Proc. 3. Internat. Conf. Spectral and High Order Methods, ICOSAHOM'95 (Houston, Texas, U.S.A., 1995), edited by A.V. Ilin and L.R. Scott (Houston Journal of Mathematics, 1996), pp. 109119.
 13.
 K. S. Eckhoff and J. H. Rolfsnes, On nonsmooth solutions of linear hyperbolic systems. J. Comp. Phys. 125, (1996), pp. 115. MR 96k:65067
 14.
 K. S. Eckhoff and C. E. Wasberg, Solution of parabolic partial differential equations in complex geometries by a modified Fourier collocation method. Proc. 3. Internat. Conf. Spectral and High Order Methods, ICOSAHOM'95 (Houston, Texas, U.S.A., 1995), edited by A.V. Ilin and L.R. Scott (Houston Journal of Mathematics, 1996), pp. 8391.
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Additional Information
Knut S. Eckhoff
Affiliation:
Department of Mathematics, University of Bergen, Johannes Bruns gate 12, N5008 Bergen\ Norway
Email:
reske@mi.uib.no
DOI:
http://dx.doi.org/10.1090/S0025571898009491
PII:
S 00255718(98)009491
Keywords:
Spectral methods,
Fourier series,
discontinuous functions,
Bernoulli polynomials,
singular functions,
quadrature,
partial differential equations
Received by editor(s):
December 11, 1996
Received by editor(s) in revised form:
March 26, 1997
Additional Notes:
This paper is partly based on work done while the author was engaged at the SINTEF Multiphase Flow Laboratory, Trondheim, Norway. The paper is also partly based on work done while the author was in residence at the Division of Applied Mathematics, Brown University, Providence, R.I., U.S.A. supported by AFOSR grant 9510074 and NSF grant DMS9500814.
Article copyright:
© Copyright 1998
American Mathematical Society
