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Order conditions for General Linear Nyström methods

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Abstract

The purpose of this paper is to analyze the algebraic theory of order for the family of general linear Nyström (GLN) methods introduced in D’Ambrosio et al. (Numer. Algorithm 61(2), 331–349, 2012) with the aim to provide a general framework for the representation and analysis of numerical methods solving initial value problems based on second order ordinary differential equations (ODEs). Our investigation is carried out by suitably extending the theory of B-series for second order ODEs to the case of GLN methods, which leads to a general set of order conditions. This allows to recover the order conditions of numerical methods already known in the literature, but also to assess a general approach to study the order conditions of new methods, simply regarding them as GLN methods: the obtained results are indeed applied to both known and new methods for second order ODEs.

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References

  1. Brouder, C.: Trees, renormalization and differential equations. BIT 44(3), 425–438 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Butcher, J.C., Tracogna, S.: Order conditions for two-step Runge-Kutta methods. Appl. Numer. Math. 24(2–3), 351–364 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Butcher, J.C., Chan, T.M.H.: A new approach to the algebraic structures for integration methods. BIT 42(3), 477–489 (2002)

    MATH  MathSciNet  Google Scholar 

  4. Butcher, J.C.: General linear methods. Acta Numer. 15, 157–256 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, Chichester (2008)

    Book  Google Scholar 

  6. Butcher, J.C.: B-series and B-series coefficients. JNAIAM J. Numer. Anal. Ind. Appl. Math. 5(1–2), 39–48 (2010)

    MathSciNet  Google Scholar 

  7. Butcher, J.C., Chan, T.M.H.: The tree and forest spaces with applications to initial-value problem methods. BIT 50(4), 713–728 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Butcher, J.C.: Trees and numerical methods for ordinary differential equations. Numer. Algorithms 53(2–3), 153–170 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chartier, P., Hairer, E., Vilmart, G.: Algebraic structures of B-series. Found. Comput. Math. 10(4), 407–427 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Coleman, J.P.: Order conditions for a class of two-step methods for \(y'' = f(x,y)\). IMA J. Numer. Anal. 23, 197–220 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cong, N.H.: Parallel-iterated pseudo two-step Runge-Kutta-Nyström methods for nonstiff second-order IVPs. Comput. Math. Appl. 44(1–2), 143–155 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Connes, A., Kreimer, D.: Hopf algebras, renormalization and noncommutative geometry. Commun. Math. Phys. 199, 203–242 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Conte, D., D’Ambrosio, R., Jackiewicz, Z., Paternoster, B.: Numerical search for algebrically stable two-step continuous Runge-Kutta methods. J. Comput. Appl. Math. 239, 304–321 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  14. D’Ambrosio, R., Ferro, M., Paternoster, B.: Two-step hybrid collocation methods for \(y''=f(x,y)\). Appl. Math. Lett. 22, 1076–1080 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. D’Ambrosio, R., Ferro, M., Jackiewicz, Z., Paternoster, B.: Two-step almost collocation methods for ordinary differential equations. Numer. Algorithms 53(2–3), 195–217 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. D’Ambrosio, R., Jackiewicz, Z.: Continuous two-step Runge–Kutta methods for ordinary differential equations. Numer. Algorithms 54(2), 169–193 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. D’Ambrosio, R., Jackiewicz, Z.: Construction and implementation of highly stable two-step continuous methods for stiff differential systems. Math. Comput. Simul. 81(9), 1707–1728 (2011)

    Article  MATH  Google Scholar 

  18. D’Ambrosio, R., Ferro, M., Paternoster, B.: Trigonometrically fitted two-step hybrid methods for second order ordinary differential equations with one or two frequencies. Math. Comput. Simul. 81, 1068–1084 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. D’Ambrosio, R., Esposito, E., Paternoster, B.: Exponentially fitted two-step hybrid methods for \(y''=f(x,y)\). J. Comput. Appl. Math. 235, 4888–4897 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. D’Ambrosio, R., Esposito, E., Paternoster, B.: General linear methods for \(y''=f(y(t))\). Numer. Algorithm 61(2), 331–349 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. D’Ambrosio, R., Esposito, E., Paternoster, B.: Parameter estimation in two-step hybrid methods for second order ordinary differential equations. J. Math. Chem. 50(1), 155–168 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. D’Ambrosio, R., Paternoster, B.: Two-step modified collocation methods with structured coefficient matrices for ordinary differential equations. Appl. Numer. Math. 62(10), 1325–1334 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hairer, E., Wanner, G.: A theory for Nyström methods. Numer. Math. 25, 383–400 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hairer, E., Wanner, G.: Order conditions for general two-step Runge-Kutta methods. SIAM J. Numer. Anal. 34(7), 2086–2089 (1997)

    MathSciNet  Google Scholar 

  25. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations, 2nd edn., Springer Series in Computational Mathematics, vol. 8. Springer, Berlin (2008)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  27. Paternoster, B.: Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials. Appl. Numer. Math. 28, 401–412 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  28. Paternoster, B.: Two step Runge-Kutta-Nystrom methods for \(y''=f(x,y)\) and P-stability. In: Sloot, P.M.A. et al. (eds.) Computational Science - ICCS 2002. Lecture Notes in Computer Science, Part III, vol. 2331, pp. 459–466. Springer, Amsterdam (2002)

  29. Paternoster, B.: Two step Runge-Kutta-Nystrom methods for oscillatory problems based on mixed polynomials. In: Sloot, P.M.A., Abramson, D., Bogdanov, A.V., Dongarra, J.J., Zomaya, A.Y., Gorbachev, Y.E. (eds.) Computational Science - ICCS 2003. Lecture Notes in Computer Science, Part II, vol. 2658, pp. 131–138. Springer, Berlin/Heidelberg (2003)

  30. Paternoster, B.: Two step Runge-Kutta-Nystrom methods based on algebraic polynomials. Rend. Mat. Appl. 23, 277–288 (2003)

    MATH  MathSciNet  Google Scholar 

  31. Paternoster, B.: A general family of two step Runge-Kutta-Nyström methods for \(y''=f(x,y)\) based on algebraic polynomials. In: Alexandrov, V.N., et al. (eds.) Computational Science - ICCS 2006. Lecture Notes in Computer Science, Part IV, vol. 3994, pp. 700–707. Springer, Amsterdam (2006)

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Authors and Affiliations

  1. University of Salerno, Fisciano, Italy

    Raffaele D’Ambrosio, Giuseppe De Martino & Beatrice Paternoster

Authors
  1. Raffaele D’Ambrosio
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  2. Giuseppe De Martino
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  3. Beatrice Paternoster
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Correspondence to Raffaele D’Ambrosio.

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Dedicated to John C. Butcher, in occasion of his 80th birthday

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D’Ambrosio, R., De Martino, G. & Paternoster, B. Order conditions for General Linear Nyström methods. Numer Algor 65, 579–595 (2014). https://doi.org/10.1007/s11075-013-9819-3

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  • DOI: https://doi.org/10.1007/s11075-013-9819-3

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