Skip to main content
Erschienen in: Numerical Algorithms 3/2020

20.12.2019 | Original Paper

Numerical methods based on the Floater–Hormann interpolants for stiff VIEs

verfasst von: Ali Abdi, Seyyed Ahmad Hosseini, Helmut Podhaisky

Erschienen in: Numerical Algorithms | Ausgabe 3/2020

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

The Floater–Hormann family of the barycentric rational interpolants has recently gained popularity because of its excellent stability properties and highly order of convergence. The purpose of this paper is to design highly accurate and stable schemes based on this family of interpolants for the numerical solution of stiff Volterra integral equations of the second kind.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Abdi, A.: General linear methods with large stability regions for Volterra integral equations. Comp. Appl. Math. 38(52), 1–16 (2019)MathSciNetMATH Abdi, A.: General linear methods with large stability regions for Volterra integral equations. Comp. Appl. Math. 38(52), 1–16 (2019)MathSciNetMATH
2.
Zurück zum Zitat Abdi, A., Berrut, J.-P., Hosseini, S. A.: The linear barycentric rational method for a class of delay Volterra integro-differential equations. J. Sci. Comput. 75, 1757–1775 (2018)MathSciNetCrossRef Abdi, A., Berrut, J.-P., Hosseini, S. A.: The linear barycentric rational method for a class of delay Volterra integro-differential equations. J. Sci. Comput. 75, 1757–1775 (2018)MathSciNetCrossRef
3.
Zurück zum Zitat Abdi, A., Fazeli, S., Hojjati, G.: Construction of efficient general linear methods for stiff Volterra integral equations. J. Comput. Appl. Math. 292, 417–429 (2016)MathSciNetCrossRef Abdi, A., Fazeli, S., Hojjati, G.: Construction of efficient general linear methods for stiff Volterra integral equations. J. Comput. Appl. Math. 292, 417–429 (2016)MathSciNetCrossRef
4.
Zurück zum Zitat Abdi, A., Hosseini, S. A.: The barycentric rational difference-quadrature scheme for systems of Volterra integro-differential equations. SIAM J. Sci. Comput. 40, A1936–A1960 (2018)MathSciNetCrossRef Abdi, A., Hosseini, S. A.: The barycentric rational difference-quadrature scheme for systems of Volterra integro-differential equations. SIAM J. Sci. Comput. 40, A1936–A1960 (2018)MathSciNetCrossRef
5.
Zurück zum Zitat Abdi, A., Hosseini, S.A., Podhaisky, H.: Adaptive linear barycentric rational finite differences method for stiff ODEs. J. Comput. Appl. Math. 357, 204–214 (2019)MathSciNetCrossRef Abdi, A., Hosseini, S.A., Podhaisky, H.: Adaptive linear barycentric rational finite differences method for stiff ODEs. J. Comput. Appl. Math. 357, 204–214 (2019)MathSciNetCrossRef
6.
Zurück zum Zitat Baker, C. T. H., Keech, M. S.: Stability regions in the numerical treatment of Volterra integral equations. SIAM J. Numer. Anal. 15, 394–417 (1978)MathSciNetCrossRef Baker, C. T. H., Keech, M. S.: Stability regions in the numerical treatment of Volterra integral equations. SIAM J. Numer. Anal. 15, 394–417 (1978)MathSciNetCrossRef
7.
Zurück zum Zitat Battles, Z., Trefethen, L. N.: An extension of Matlab to continuous functions and operators. SIAM J. Sci. Comput. 25, 1743–1770 (2004)MathSciNetCrossRef Battles, Z., Trefethen, L. N.: An extension of Matlab to continuous functions and operators. SIAM J. Sci. Comput. 25, 1743–1770 (2004)MathSciNetCrossRef
8.
Zurück zum Zitat Berrut, J. -P.: Rational functions for guaranteed and experimentally well-conditioned global interpolation. Comput. Math. Appl. 15, 1–16 (1988)MathSciNetCrossRef Berrut, J. -P.: Rational functions for guaranteed and experimentally well-conditioned global interpolation. Comput. Math. Appl. 15, 1–16 (1988)MathSciNetCrossRef
9.
Zurück zum Zitat Berrut, J. -P.: Linear barycentric rational interpolation with guaranteed degree of exactness. In: Fasshauer, G.E., Schumaker, L.L. (eds.) Approximation Theory XV: San Antonio 2016, Springer Proceedings in Mathematics & Statistics, 1–20 (2017) Berrut, J. -P.: Linear barycentric rational interpolation with guaranteed degree of exactness. In: Fasshauer, G.E., Schumaker, L.L. (eds.) Approximation Theory XV: San Antonio 2016, Springer Proceedings in Mathematics & Statistics, 1–20 (2017)
10.
Zurück zum Zitat Berrut, J. -P., Hosseini, S. A., Klein, G.: The linear barycentric rational quadrature method for Volterra integral equations. SIAM J. Sci. Comput. 36, A105–A123 (2014)MathSciNetCrossRef Berrut, J. -P., Hosseini, S. A., Klein, G.: The linear barycentric rational quadrature method for Volterra integral equations. SIAM J. Sci. Comput. 36, A105–A123 (2014)MathSciNetCrossRef
11.
12.
13.
Zurück zum Zitat Blom, J. G., Brunner, H.: The numerical solution of nonlinear Volterra integral equations of the second kind by collocation and iterated collocation methods. SIAM J. Sci. Stat. Comput. 8, 806–830 (1987)MathSciNetCrossRef Blom, J. G., Brunner, H.: The numerical solution of nonlinear Volterra integral equations of the second kind by collocation and iterated collocation methods. SIAM J. Sci. Stat. Comput. 8, 806–830 (1987)MathSciNetCrossRef
14.
Zurück zum Zitat Brunner, H.: Collocation methods for Volterra integral and related functional equations. Cambridge University Press, Cambridge (2004)CrossRef Brunner, H.: Collocation methods for Volterra integral and related functional equations. Cambridge University Press, Cambridge (2004)CrossRef
15.
Zurück zum Zitat Brunner, H., Nørsett, S.P., Wolkenfelt, P.H.M.: On V0-stability of numerical methods for Volterra integral equations of the second kind. Report NW84/80. Mathematish Centrum, Amsterdam (1980) Brunner, H., Nørsett, S.P., Wolkenfelt, P.H.M.: On V0-stability of numerical methods for Volterra integral equations of the second kind. Report NW84/80. Mathematish Centrum, Amsterdam (1980)
16.
Zurück zum Zitat Brunner, H., van der Houwen, P. J.: The numerical solution of Volterra equations. CWI Monographs, North-Holland (1986)MATH Brunner, H., van der Houwen, P. J.: The numerical solution of Volterra equations. CWI Monographs, North-Holland (1986)MATH
17.
Zurück zum Zitat Capobianco, G., Conte, D., Del Prete, I., Russo, E.: Fast Runge–Kutta methods for nonlinear convolution systems of Volterra integral equations. BIT 47, 259–275 (2007)MathSciNetCrossRef Capobianco, G., Conte, D., Del Prete, I., Russo, E.: Fast Runge–Kutta methods for nonlinear convolution systems of Volterra integral equations. BIT 47, 259–275 (2007)MathSciNetCrossRef
18.
Zurück zum Zitat Conte, D., Jackiewicz, Z., Paternoster, B.: Two-step almost collocation methods for Volterra integral equations. Appl. Math. Comput. 204, 839–853 (2008)MathSciNetMATH Conte, D., Jackiewicz, Z., Paternoster, B.: Two-step almost collocation methods for Volterra integral equations. Appl. Math. Comput. 204, 839–853 (2008)MathSciNetMATH
19.
Zurück zum Zitat Conte, D., Paternoster, B.: Multistep collocation methods for Volterra integral equations. Appl. Numer. Math. 59, 1721–1736 (2009)MathSciNetCrossRef Conte, D., Paternoster, B.: Multistep collocation methods for Volterra integral equations. Appl. Numer. Math. 59, 1721–1736 (2009)MathSciNetCrossRef
20.
Zurück zum Zitat Floater, M. S., Hormann, K.: Barycentric rational interpolation with no poles and high rates of approximation. Numer. Math. 107, 315–331 (2007)MathSciNetCrossRef Floater, M. S., Hormann, K.: Barycentric rational interpolation with no poles and high rates of approximation. Numer. Math. 107, 315–331 (2007)MathSciNetCrossRef
21.
Zurück zum Zitat Guttel, S., Klein, G.: Convergence of linear barycentric rational interpolation for analytic functions. SIAM J. Numer. Anal. 50, 2560–2580 (2012)MathSciNetCrossRef Guttel, S., Klein, G.: Convergence of linear barycentric rational interpolation for analytic functions. SIAM J. Numer. Anal. 50, 2560–2580 (2012)MathSciNetCrossRef
22.
Zurück zum Zitat Hairer, E., Lubich, C., Schlichte, M.: Fast numerical solution of nonlinear Volterra convolution equations. SIAM J. Sci. Stat. Comput. 6, 532–541 (1985)MathSciNetCrossRef Hairer, E., Lubich, C., Schlichte, M.: Fast numerical solution of nonlinear Volterra convolution equations. SIAM J. Sci. Stat. Comput. 6, 532–541 (1985)MathSciNetCrossRef
23.
Zurück zum Zitat Henrici, P.: Essentials of Numerical Analysis. John Wiley, New York (1982)MATH Henrici, P.: Essentials of Numerical Analysis. John Wiley, New York (1982)MATH
24.
Zurück zum Zitat Hetcote, H. W., Tudor, D. W.: Integral equation models for endemic infectious diseases. J. Math. Biol. 9, 37–47 (1980)MathSciNetCrossRef Hetcote, H. W., Tudor, D. W.: Integral equation models for endemic infectious diseases. J. Math. Biol. 9, 37–47 (1980)MathSciNetCrossRef
25.
Zurück zum Zitat Hoppensteadt, F. C., Jackiewicz, Z., Zubik-Kowal, B.: Numerical solution of Volterra integral and integro-differential equations with rapidly vanishing convolution kernels. BIT 47, 325–350 (2007)MathSciNetCrossRef Hoppensteadt, F. C., Jackiewicz, Z., Zubik-Kowal, B.: Numerical solution of Volterra integral and integro-differential equations with rapidly vanishing convolution kernels. BIT 47, 325–350 (2007)MathSciNetCrossRef
26.
Zurück zum Zitat Hosseini, S. A., Abdi, A.: On the numerical stability of the linear barycentric rational quadrature method for Volterra integral equations. Appl. Numer. Math. 100, 1–13 (2016)MathSciNetCrossRef Hosseini, S. A., Abdi, A.: On the numerical stability of the linear barycentric rational quadrature method for Volterra integral equations. Appl. Numer. Math. 100, 1–13 (2016)MathSciNetCrossRef
27.
Zurück zum Zitat Izzo, G., Jackiewicz, Z., Messina, E., Vecchio, A.: General linear methods for Volterra integral equations. J. Comput. Appl. Math. 234, 2768–2782 (2010)MathSciNetCrossRef Izzo, G., Jackiewicz, Z., Messina, E., Vecchio, A.: General linear methods for Volterra integral equations. J. Comput. Appl. Math. 234, 2768–2782 (2010)MathSciNetCrossRef
28.
Zurück zum Zitat Izzo, G., Russo, E., Chiapparelli, C.: Highly stable Runge–Kutta methods for Volterra integral equations. Appl. Numer. Math. 62, 1002–1013 (2012)MathSciNetCrossRef Izzo, G., Russo, E., Chiapparelli, C.: Highly stable Runge–Kutta methods for Volterra integral equations. Appl. Numer. Math. 62, 1002–1013 (2012)MathSciNetCrossRef
29.
Zurück zum Zitat Klein, G.: Applications of Linear Barycentric Rational Interpolation. University of Fribourg, PhD thesis (2012)MATH Klein, G.: Applications of Linear Barycentric Rational Interpolation. University of Fribourg, PhD thesis (2012)MATH
31.
Zurück zum Zitat Klein, G., Berrut, J. -P.: Linear rational finite differences from derivatives of barycentric rational interpolants. SIAM J. Numer. Anal. 50, 643–656 (2012)MathSciNetCrossRef Klein, G., Berrut, J. -P.: Linear rational finite differences from derivatives of barycentric rational interpolants. SIAM J. Numer. Anal. 50, 643–656 (2012)MathSciNetCrossRef
32.
Zurück zum Zitat Linz, P.: Analytical and Numerical Methods for Volterra Equations. SIAM, Philadelphia (1985)CrossRef Linz, P.: Analytical and Numerical Methods for Volterra Equations. SIAM, Philadelphia (1985)CrossRef
33.
Zurück zum Zitat Tang, T., Xu, X., Cheng, J.: On spectral methods for Volterra integral equations and the convergence analysis. J. Comput. Math. 26, 825–837 (2008)MathSciNetMATH Tang, T., Xu, X., Cheng, J.: On spectral methods for Volterra integral equations and the convergence analysis. J. Comput. Math. 26, 825–837 (2008)MathSciNetMATH
36.
Zurück zum Zitat van der Houwen, P. J., te Riele, H. J. J.: Backward differentiation type formulas for Volterra integral equations of the second kind. Numer. Math. 37, 205–217 (1981)MathSciNetCrossRef van der Houwen, P. J., te Riele, H. J. J.: Backward differentiation type formulas for Volterra integral equations of the second kind. Numer. Math. 37, 205–217 (1981)MathSciNetCrossRef
Metadaten
Titel
Numerical methods based on the Floater–Hormann interpolants for stiff VIEs
verfasst von
Ali Abdi
Seyyed Ahmad Hosseini
Helmut Podhaisky
Publikationsdatum
20.12.2019
Verlag
Springer US
Erschienen in
Numerical Algorithms / Ausgabe 3/2020
Print ISSN: 1017-1398
Elektronische ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-019-00841-4

Weitere Artikel der Ausgabe 3/2020

Numerical Algorithms 3/2020 Zur Ausgabe

Premium Partner