On a high order numerical method for

functions with singularities

Author:
Knut S. Eckhoff

Journal:
Math. Comp. **67** (1998), 1063-1087

MSC (1991):
Primary 65M70, 65N35

MathSciNet review:
1459387

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**1.**Tom M. Apostol,*Calculus. Vol. II: Multi-variable calculus and linear algebra, with applications to differential equations and probability*, Second edition, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1969. MR**0248290****2.**E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen,*LAPACK Users' Guide,*Society for Industrial and Applied Mathematics, Philadelphia, PA, (1992).**3.**Carl M. Bender and Steven A. Orszag,*Advanced mathematical methods for scientists and engineers*, McGraw-Hill Book Co., New York, 1978. International Series in Pure and Applied Mathematics. MR**538168****4.**J. P. Boyd,*Chebyshev & Fourier Spectral Methods,*Lecture Notes in Engineering**49**, Springer-Verlag, Berlin, (1989).**5.**Kenneth P. Bube,*𝐶^{𝑚} convergence of trigonometric interpolants*, SIAM J. Numer. Anal.**15**(1978), no. 6, 1258–1268. MR**512698**, 10.1137/0715086**6.**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****7.**Philip J. Davis and Philip Rabinowitz,*Methods of numerical integration*, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR**760629****8.**Knut S. Eckhoff,*Accurate and efficient reconstruction of discontinuous functions from truncated series expansions*, Math. Comp.**61**(1993), no. 204, 745–763. MR**1195430**, 10.1090/S0025-5718-1993-1195430-1**9.**Knut S. Eckhoff,*On discontinuous solutions of hyperbolic equations*, Comput. Methods Appl. Mech. Engrg.**116**(1994), no. 1-4, 103–112. ICOSAHOM’92 (Montpellier, 1992). MR**1286518**, 10.1016/S0045-7825(94)80013-8**10.**Knut S. Eckhoff,*Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions*, Math. Comp.**64**(1995), no. 210, 671–690. MR**1265014**, 10.1090/S0025-5718-1995-1265014-7**11.**K. S. Eckhoff,*On a high order numerical method for solving partial differential equations in complex geometries.*J. Scient. Comp.**12**(1997), pp. 119-138.**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. 109-119.**13.**Knut S. Eckhoff and Jens H. Rolfsnes,*On nonsmooth solutions of linear hyperbolic systems*, J. Comput. Phys.**125**(1996), no. 1, 1–15. MR**1381801**, 10.1006/jcph.1996.0075**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. 83-91.**15.**Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,*Higher transcendental functions. Vols. I, II*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR**0058756****16.**Gene H. Golub and Charles F. Van Loan,*Matrix computations*, 2nd ed., Johns Hopkins Series in the Mathematical Sciences, vol. 3, Johns Hopkins University Press, Baltimore, MD, 1989. MR**1002570****17.**David Gottlieb and Steven A. Orszag,*Numerical analysis of spectral methods: theory and applications*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26. MR**0520152****18.**L. V. Kantorovich and V. I. Krylov,*Approximate methods of higher analysis*, Translated from the 3rd Russian edition by C. D. Benster, Interscience Publishers, Inc., New York; P. Noordhoff Ltd., Groningen, 1958. MR**0106537****19.**Heinz-Otto Kreiss and Joseph Oliger,*Stability of the Fourier method*, SIAM J. Numer. Anal.**16**(1979), no. 3, 421–433. MR**530479**, 10.1137/0716035**20.**Cornelius Lanczos,*Discourse on Fourier series*, Hafner Publishing Co., New York, 1966. MR**0199629****21.**J. N. Lyness,*Computational techniques based on the Lanczos representation*, Math. Comp.**28**(1974), 81–123. MR**0334458**, 10.1090/S0025-5718-1974-0334458-6**22.**Gilbert Strang and George J. Fix,*An analysis of the finite element method*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. MR**0443377****23.**Hervé Vandeven,*Family of spectral filters for discontinuous problems*, J. Sci. Comput.**6**(1991), no. 2, 159–192. MR**1140344**, 10.1007/BF01062118**24.**A. Zygmund,*Trigonometric series: Vols. I, II*, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968. MR**0236587**

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
65M70,
65N35

Retrieve articles in all journals with MSC (1991): 65M70, 65N35

Additional Information

**Knut S. Eckhoff**

Affiliation:
Department of Mathematics, University of Bergen, Johannes Bruns gate 12, N-5008 Bergen Norway

Email:
reske@mi.uib.no

DOI:
https://doi.org/10.1090/S0025-5718-98-00949-1

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 95-1-0074 and NSF grant DMS-9500814.

Article copyright:
© Copyright 1998
American Mathematical Society