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Natural Volterra Runge-Kutta methods

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Abstract

A very general class of Runge-Kutta methods for Volterra integral equations of the second kind is analyzed. Order and stage order conditions are derived for methods of order p and stage order q = p up to the order four. We also investigate stability properties of these methods with respect to the basic and the convolution test equations. The systematic search for A- and V 0-stable methods is described and examples of highly stable methods are presented up to the order p = 4 and stage order q = 4.

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References

  1. Albrecht, P.: A new theoretical approach to Runge-Kutta methods. SIAM J. Numer. Anal. 24, 391–406 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Albrecht, P.: Runge-Kutta theory in a nutshell. SIAM J. Numer. Anal. 33, 1712–1735 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Amini, S.: Stability analysis of methods employing reducible rules for Volterra integral equations. BIT 23, 322–328 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bellen, A., Jackiewicz, Z., Vermiglio, R., Zennaro, M.: Natural continuous extensions of Runge-Kutta methods for Volterra integral equations of the second kind and their applications. Math. Comput. 52, 49–63 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bellen, A., Jackiewicz, Z., Vermiglio, R., Zennaro, M.: Stability analysis of Runge-Kutta methods for Volterra integral equations of the second kind. IMA J. Numer. Anal. 10, 103–118 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bel’tyukov, B.A.: An analogue of the Runge-Kutta method for the solution of nonlinear integral equations of Volterra type. Diff. Equat. 1, 417–433 (1965)

    Google Scholar 

  7. Brunner, H., Hairer, E., Nørsett, S.P.: Runge-Kutta theory for Volterra integral equations of the second kind. Math. Comput. 39, 147–163 (1982)

    Article  MATH  Google Scholar 

  8. Brunner, H., van der Houwen, P.J.: The Numerical Solution of Volterra Equations. CWI Monographs 3. North-Holland, Amsterdam (1986)

    Google Scholar 

  9. Brunner, H., Nørsett, S.P., Wolkenfelt, P.H.M.: On V 0-stability of numerical methods for Volterra integral equations of the second kind. Report NW 84/80. Mathematisch Centrum, Amsterdam (1980)

    Google Scholar 

  10. Cardone, A., Conte, D.: Multistep collocation methods for Volterra integro-differential equations. Appl. Math. Comput. 221, 770–785 (2013)

    Article  MathSciNet  Google Scholar 

  11. Conte, D., Jackiewicz, Z., Paternoster, B.: Two-step almost collocation methods for Volterra integral equations. Appl. Math. Comp. 204, 839–853 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Conte, D., Paternoster, B.: Multistep collocation methods for Volterra integral equations. Appl. Numer. Math. 59, 1721–1736 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Conte, D., D’Ambrosio, R., Paternoster, B.: Two-step diagonally-implicit collocation based methods for Volterra integral equations. Appl. Numer. Math. 62, 1312–1324 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Garrappa, R.: Order conditions for Volterra Runge-Kutta methods. Appl. Numer. Math. 60, 561–573 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hairer, E.: Order conditions for numerical methods for partitioned ordinary differential equations. Numer. Math. 36, 431–445 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer, New York (1996)

    Book  MATH  Google Scholar 

  17. van der Houwen, P.J.: Convergence and stability results in Runge-Kutta methods for Volterra integral equations of the second kind. BIT 20, 375–377 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  18. van der Houwen, P.J., Wolkenfelt, P.J., Baker, C.T.H.: Convergence and stability analysis for modified Runge-Kutta methods in the numerical treatment of second-kind Volterra integral equations. IMA J. Numer. Anal. 1, 303–328 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  19. Izzo, G., Jackiewicz, Z., Messina, E., Vecchio, A.: General linear methods for Volterra integral equations. J. Comput. Appl. Math. 234, 2768–2782 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Izzo, G., Russo, E., Chiapparelli, C.: Highly stable Runge-Kutta methods for Volterra integral equations. Appl. Numer. Math. 62, 1001–1013 (2012)

    Article  MathSciNet  Google Scholar 

  21. Jackiewicz, Z.: General Linear Methods for Ordinary Differential Equations. Wiley, Hoboken (2009)

    Book  MATH  Google Scholar 

  22. Jackiewicz, Z., Tracogna, S.: A general class of two-step Runge-Kutta methods for ordinary differential equations. SIAM J. Numer. Anal. 32, 1390–1427 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  23. Lambert, J.D.: Computational Methods in Ordinary Differential Equations. Wiley, Chichester (1973)

    MATH  Google Scholar 

  24. Pouzet, P.: Etude en vue de leur traitement numérique des équations intégrales de type Volterra. Rev. Français Traitement Information (Chiffres) 6, 79–112 (1963)

    MathSciNet  Google Scholar 

  25. Schur, J.: Über Potenzreihen die im Innern des Einheitskreises beschrankt sind. J. Reine Angew. Math. 147, 205–232 (1916)

    Google Scholar 

  26. Wolkenfelt, P.H.M.: On the numerical stability of reducible quadrature methods for second kind Volterra integral equations. Z. Angew. Math. Mech. 61, 399–401 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  27. Zennaro, M.: Natural continuous extensions of Runge-Kutta methods. Math. Comput. 46, 119–133 (1986)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. University of Salerno, Fisciano, Italy

    Dajana Conte & Raffaele D’Ambrosio

  2. University of Naples “Federico II”, Naples, Italy

    Giuseppe Izzo

  3. Arizona State University, Tempe, USA

    Zdzislaw Jackiewicz

  4. AGH University of Science and Technology, Kraków, Poland

    Zdzislaw Jackiewicz

Authors
  1. Dajana Conte
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  2. Raffaele D’Ambrosio
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  3. Giuseppe Izzo
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  4. Zdzislaw Jackiewicz
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Corresponding author

Correspondence to Giuseppe Izzo.

Additional information

Partially supported by the School of Sciences and Technology - University of Naples “Federico II” under the project “F.A.R.O.” Control and stability of diffusive processes in the environment.

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Conte, D., D’Ambrosio, R., Izzo, G. et al. Natural Volterra Runge-Kutta methods. Numer Algor 65, 421–445 (2014). https://doi.org/10.1007/s11075-013-9790-z

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  • DOI: https://doi.org/10.1007/s11075-013-9790-z

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