research-article

Accuracy and Speed in Computing the Chebyshev Collocation Derivative

Authors:

Wai Sun Don,

Alex SolomonoffAuthors Info & Claims

SIAM Journal on Scientific Computing, Volume 16, Issue 6

Pages 1253 - 1268

https://doi.org/10.1137/0916073

Published: 01 November 1995 Publication History

Abstract

We study several algorithms for computing the Chebyshev spectral derivative and compare their roundoff error. For a large number of collocation points, the elements of the Chebyshev differentiation matrix, if constructed in the usual way, are not computed accurately. A subtle cause is found to account for the poor accuracy when computing the derivative by the matrix-vector multiplication method. Methods for accurately computing the elements of the matrix are presented and we find that if the entries of the matrix are computed accurately, the roundoff error of the matrix-vector multiplication is as small as that of the transform-recursion algorithm.

Furthermore, results of the CPU time usage are shown for several different algorithms for computing the derivative by the Chebyshev collocation method for a wide variety of two-dimensional grid sizes on both an IBM mainframe and a Cray 2 computer. We find that which algorithm is fastest on a particular machine depends not only on the grid size, but also on small details of the computer hardware as well. For most practical grid sizes used in computation, the even-odd decomposition algorithm is found to be faster than the transform-recursion method.

References

[1]

David Gottlieb, S. Orszag, Numerical analysis of spectral methods: theory and applications, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977v+172

Crossref

Google Scholar

[2]

C. Canuto, A. Quarteroni, M. Y. Hussaini, T. Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988xiv+557

Crossref

Google Scholar

[3]

K. Breuer, R. Everson, On the errors incurred calculating derivatives using Chebyshev polynomials, J. Comput. Phys., 99 (1992), 56–67

Digital Library

Google Scholar

[4]

E. Rothman, M. Durand, F. El Dabaghi, Reducing round-of, error in Chebyshev pseudospectral computations, Proc. 2nd Symposium on High-Performance Computing, Montpellier, France, 7-9 October 1991, Elsevier Science Publishers, New York, 1992

Google Scholar

[5]

Alex Solomonoff, A fast algorithm for spectral differentiation, J. Comput. Phys., 98 (1992), 174–177

Digital Library

Google Scholar

[6]

Lloyd N. Trefethen, Manfred R. Trummer, An instability phenomenon in spectral methods, SIAM J. Numer. Anal., 24 (1987), 1008–1023

Digital Library

Google Scholar

Cited By

View all

Fang YDu LHe CSun DYang LFu QSun X(2024)BiGlobal stability analysis for flow in complex geometry based on immersed boundary methodJournal of Computational Physics10.1016/j.jcp.2023.112630497:COnline publication date: 15-Jan-2024
https://dl.acm.org/doi/10.1016/j.jcp.2023.112630
Świrydowicz KChalmers NKarakus AWarburton T(2019)Acceleration of tensor-product operations for high-order finite element methodsInternational Journal of High Performance Computing Applications10.1177/109434201881636833:4(735-757)Online publication date: 1-Jul-2019
https://dl.acm.org/doi/10.1177/1094342018816368
Kheybari SDarvishi MHashemi M(2019)Numerical simulation for the space-fractional diffusion equationsApplied Mathematics and Computation10.1016/j.amc.2018.11.041348:C(57-69)Online publication date: 1-May-2019
https://dl.acm.org/doi/10.1016/j.amc.2018.11.041
Show More Cited By

Index Terms

Accuracy and Speed in Computing the Chebyshev Collocation Derivative
1. Mathematics of computing
  1. Mathematical analysis
    1. Functional analysis
      1. Approximation
    2. Numerical analysis
      1. Computations on matrices
      2. Numerical differentiation
2. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis

Index terms have been assigned to the content through auto-classification.

Recommendations

Accuracy Enhancement for Higher Derivatives using Chebyshev Collocation and a Mapping Technique

A new method is investigated to reduce the roundoff error in computing derivatives using Chebyshev collocation methods. By using a grid mapping derived by Kosloff and Tal-Ezer, and the proper choice of the parameter $\alpha$, the roundoff error of the kth ...
Integration Preconditioning Matrix for Ultraspherical Pseudospectral Operators

In this paper, we give an integration rectangular preconditioner for ultraspherical pseudospectral operators. We present an explicit solution of matrix equations with the structure provided by the pseudospectral discretization of the nth-order ...
A Chebyshev spectral collocation method for solving Burgers'-type equations

In this paper, we elaborated a spectral collocation method based on differentiated Chebyshev polynomials to obtain numerical solutions for some different kinds of nonlinear partial differential equations. The problem is reduced to a system of ordinary ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image SIAM Journal on Scientific Computing

SIAM Journal on Scientific Computing Volume 16, Issue 6

Nov 1995

271 pages

ISSN:1064-8275

Issue’s Table of Contents

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 November 1995

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

26
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 05 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

View all

Fang YDu LHe CSun DYang LFu QSun X(2024)BiGlobal stability analysis for flow in complex geometry based on immersed boundary methodJournal of Computational Physics10.1016/j.jcp.2023.112630497:COnline publication date: 15-Jan-2024
https://dl.acm.org/doi/10.1016/j.jcp.2023.112630
Świrydowicz KChalmers NKarakus AWarburton T(2019)Acceleration of tensor-product operations for high-order finite element methodsInternational Journal of High Performance Computing Applications10.1177/109434201881636833:4(735-757)Online publication date: 1-Jul-2019
https://dl.acm.org/doi/10.1177/1094342018816368
Kheybari SDarvishi MHashemi M(2019)Numerical simulation for the space-fractional diffusion equationsApplied Mathematics and Computation10.1016/j.amc.2018.11.041348:C(57-69)Online publication date: 1-May-2019
https://dl.acm.org/doi/10.1016/j.amc.2018.11.041
Moore M(2017)A fast Chebyshev method for simulating flexible-wing propulsionJournal of Computational Physics10.1016/j.jcp.2017.05.052345:C(792-817)Online publication date: 15-Sep-2017
https://dl.acm.org/doi/10.1016/j.jcp.2017.05.052
Xu K(2016)The Chebyshev points of the first kindApplied Numerical Mathematics10.1016/j.apnum.2015.12.002102:C(17-30)Online publication date: 1-Apr-2016
https://dl.acm.org/doi/10.1016/j.apnum.2015.12.002
Motsa S(2014)On the practical use of the spectral homotopy analysis method and local linearisation method for unsteady boundary-layer flows caused by an impulsively stretching plateNumerical Algorithms10.1007/s11075-013-9766-z66:4(865-883)Online publication date: 1-Aug-2014
https://dl.acm.org/doi/10.1007/s11075-013-9766-z
Motsa SShateyi SMarewo GSibanda P(2012)An improved spectral homotopy analysis method for MHD flow in a semi-porous channelNumerical Algorithms10.1007/s11075-011-9523-060:3(463-481)Online publication date: 1-Jul-2012
https://dl.acm.org/doi/10.1007/s11075-011-9523-0
Sibanda PKhidir A(2011)A new modification of the HPM for the duffing equation with cubic nonlinearityProceedings of the 2011 international conference on Applied & computational mathematics10.5555/2001305.2001324(139-143)Online publication date: 27-May-2011
https://dl.acm.org/doi/10.5555/2001305.2001324
Javidi M(2011)A modified Chebyshev pseudospectral DD algorithm for the GBH equationComputers & Mathematics with Applications10.1016/j.camwa.2011.08.05162:9(3366-3377)Online publication date: 1-Nov-2011
https://dl.acm.org/doi/10.1016/j.camwa.2011.08.051
Motsa SSibanda P(2010)A new algorithm for solving singular IVPs of Lane-Emden typeProceedings of the 4th international conference on Applied mathematics, simulation, modelling10.5555/1895214.1895244(176-180)Online publication date: 22-Jul-2010
https://dl.acm.org/doi/10.5555/1895214.1895244
Show More Cited By

Abstract

References

Cited By

Index Terms

Recommendations

Accuracy Enhancement for Higher Derivatives using Chebyshev Collocation and a Mapping Technique

Integration Preconditioning Matrix for Ultraspherical Pseudospectral Operators

A Chebyshev spectral collocation method for solving Burgers'-type equations

Comments

Information

Published In

Publisher

Publication History

Author Tags

Author Tags

Qualifiers

Contributors

Other Metrics

Bibliometrics

Article Metrics

Other Metrics

Citations

Cited By

View options

Figures

Other

Share

Share this Publication link

Share on social media

Affiliations