## On a high order numerical method for functions with singularities

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- by Knut S. Eckhoff PDF
- Math. Comp.
**67**(1998), 1063-1087 Request permission

## 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.## References

- Tom M. Apostol,
*Calculus. Vol. II: Multi-variable calculus and linear algebra, with applications to differential equations and probability*, 2nd ed., Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1969. MR**0248290** - 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). - Carl M. Bender and Steven A. Orszag,
*Advanced mathematical methods for scientists and engineers*, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. MR**538168** - J. P. Boyd,
*Chebyshev & Fourier Spectral Methods,*Lecture Notes in Engineering**49**, Springer-Verlag, Berlin, (1989). - Kenneth P. Bube,
*$C^{m}$ convergence of trigonometric interpolants*, SIAM J. Numer. Anal.**15**(1978), no. 6, 1258–1268. MR**512698**, DOI 10.1137/0715086 - 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 - 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** - Knut S. Eckhoff,
*Accurate and efficient reconstruction of discontinuous functions from truncated series expansions*, Math. Comp.**61**(1993), no. 204, 745–763. MR**1195430**, DOI 10.1090/S0025-5718-1993-1195430-1 - 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**, DOI 10.1016/S0045-7825(94)80013-8 - 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**, DOI 10.1090/S0025-5718-1995-1265014-7 - 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. - 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. - 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**, DOI 10.1006/jcph.1996.0075 - 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. - Sam Perlis,
*Maximal orders in rational cyclic algebras of composite degree*, Trans. Amer. Math. Soc.**46**(1939), 82–96. MR**15**, DOI 10.1090/S0002-9947-1939-0000015-X - 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** - David Gottlieb and Steven A. Orszag,
*Numerical analysis of spectral methods: theory and applications*, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. MR**0520152**, DOI 10.1137/1.9781611970425 - L. V. Kantorovich and V. I. Krylov,
*Approximate methods of higher analysis*, Interscience Publishers, Inc., New York; P. Noordhoff Ltd., Groningen 1958. Translated from the 3rd Russian edition by C. D. Benster. MR**0106537** - Heinz-Otto Kreiss and Joseph Oliger,
*Stability of the Fourier method*, SIAM J. Numer. Anal.**16**(1979), no. 3, 421–433. MR**530479**, DOI 10.1137/0716035 - Cornelius Lanczos,
*Discourse on Fourier series*, Hafner Publishing Co., New York, 1966. MR**0199629** - J. N. Lyness,
*Computational techniques based on the Lanczos representation*, Math. Comp.**28**(1974), 81–123. MR**334458**, DOI 10.1090/S0025-5718-1974-0334458-6 - Gilbert Strang and George J. Fix,
*An analysis of the finite element method*, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR**0443377** - Hervé Vandeven,
*Family of spectral filters for discontinuous problems*, J. Sci. Comput.**6**(1991), no. 2, 159–192. MR**1140344**, DOI 10.1007/BF01062118 - A. Zygmund,
*Trigonometric series: Vols. I, II*, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR**0236587**

## 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
- 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.
- © Copyright 1998 American Mathematical Society
- Journal: Math. Comp.
**67**(1998), 1063-1087 - MSC (1991): Primary 65M70, 65N35
- DOI: https://doi.org/10.1090/S0025-5718-98-00949-1
- MathSciNet review: 1459387