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Continuous two-step Runge–Kutta methods for ordinary differential equations

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Abstract

New classes of continuous two-step Runge-Kutta methods for the numerical solution of ordinary differential equations are derived. These methods are developed imposing some interpolation and collocation conditions, in order to obtain desirable stability properties such as A-stability and L-stability. Particular structures of the stability polynomial are also investigated.

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References

  1. Bartoszewski, Z., Jackiewicz, Z.: Construction of two-step Runge-Kutta methods of high order for ordinary differential equations. Numer. Algorithms 18, 51–70 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bartoszewski, Z., Jackiewicz, Z.: Stability analysis of two-step Runge-Kutta methods for delay differential equations. Comput. Math. Appl. 44, 83–93 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bartoszewski, Z., Jackiewicz, Z.: Nordsieck representation of two-step Runge-Kutta methods for ordinary differential equations. Appl. Numer. Math. 53, 149–163 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bartoszewski, Z., Jackiewicz, Z.: Derivation of continuous explicit two-step Runge-Kutta methods of order three. J. Comput. Appl. Math. 205, 764–776 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bartoszewski, Z., Jackiewicz, Z.: Construction of highly stable parallel two-step Runge-Kutta methods for delay differential equations. J. Comput. Appl. Math. 1–2(220), 257–270 (2008)

    Article  MathSciNet  Google Scholar 

  6. Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods, xvi+512 pp. Wiley, Chichester (1987)

    MATH  Google Scholar 

  7. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn., xx+463 pp. Wiley, Chichester (2008)

    MATH  Google Scholar 

  8. Butcher, J.C., Tracogna, S.: Order conditions for two-step Runge-Kutta methods. Appl. Numer. Math. 24, 351–364 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chollom J., Jackiewicz Z.: Construction of two-step Runge-Kutta methods with large regions of absolute stability. J. Comput. Appl. Math. 157, 125–137 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. D’Ambrosio, R.: Two-step collocation methods for ordinary differential equations. Ph.D. thesis, University of Salerno (2009)

  12. D’Ambrosio, R., Ferro, M., Jackiewicz, Z., Paternoster, B.: Two-step almost collocation methods for ordinary differential equations. Numer. Algorithms. doi:10.1007/s11075-009-9280-5

  13. D’Ambrosio, R., Ferro, M., Paternoster, B.: Collocation based two step Runge–Kutta methods for ordinary differential equations. In: Gervasi, O., et al. (eds.) ICCSA 2008. Lecture Notes in Comput. Sci., Part II, vol. 5073, pp. 736–751, Springer, New York (2008)

    Google Scholar 

  14. Guillou, A., Soulé, F.L.: La résolution numérique des problèmes differentiels aux conditions par des méthodes de collocation. RAIRO Anal. Numér. Ser. Rouge v. R-3, 17–44 (1969)

    Google Scholar 

  15. Hairer, E., Wanner, G.: Order conditions for general two-step Runge-Kutta methods. SIAM J. Numer. Anal. 34, 2087–2089 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II—Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics, vol. 14. Springer, Berlin (2002)

    Google Scholar 

  17. Jackiewicz, Z.: General Linear Methods for Ordinary Differential Equations. Wiley, New York (2009)

    Book  Google Scholar 

  18. 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 

  19. Jackiewicz, Z., Tracogna, S.: Variable stepsize continuous two-step Runge-Kutta methods for ordinary differential equations. Numer. Algorithms 12, 347–368 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  20. Jackiewicz, Z., Verner, J.H.: Derivation and implementation of two-step Runge-Kutta pairs. Jpn. J. Ind. Appl. Math. 19, 227–248 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lambert, J.D.: Computational Methods in Ordinary Differential Equations, xv+278 pp. Wiley, Chichester (1973)

    MATH  Google Scholar 

  22. Lie, I., Norsett, S.P.: Superconvergence for multistep collocation. Math. Comput. 52(185), 65–79 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  23. Prothero, A., Robinson, A.: On the stability and the accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comput. 28, 145–162 (1974)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  25. Tracogna, S.: Implementation of two-step Runge-Kutta methods for ordinary differential equations. J. Comput. Appl. Math. 76, 113–136 (1997)

    Article  MathSciNet  Google Scholar 

  26. Tracogna, S., Welfert, B.: Two-step Runge-Kutta: theory and practice. BIT 40, 775–799 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Universitá degli Studi di Salerno, Fisciano (SA), 84084, Italy

    Raffaele D’Ambrosio

  2. Department of Mathematics and Statistics, Arizona State University, Tempe, AZ, 85287, USA

    Zdzislaw Jackiewicz

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

    Zdzislaw Jackiewicz

Authors
  1. Raffaele D’Ambrosio
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  2. Zdzislaw Jackiewicz
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Correspondence to Raffaele D’Ambrosio.

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D’Ambrosio, R., Jackiewicz, Z. Continuous two-step Runge–Kutta methods for ordinary differential equations. Numer Algor 54, 169–193 (2010). https://doi.org/10.1007/s11075-009-9329-5

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  • DOI: https://doi.org/10.1007/s11075-009-9329-5

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