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Efficient Nordsieck second derivative general linear methods: construction and implementation

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Abstract

In this paper, by considering the order conditions for second derivative general linear methods of order p and stage order \(q=p-1\), we investigate construction and implementation of these methods in the Nordsieck form with \(r=s+1=p\), where s and r are the number of internal and external stages of the method, respectively. Constructed methods are A- and L-stable which possess Runge–Kutta stability property. Some numerical experiments are provided in a variable stepsize environment to validate the efficiency of the constructed methods and reliability of the proposed error estimates.

Keywords

Stiff differential equations General linear methods Second derivative methods Runge–Kutta stability A- and L-stability Variable stepsize 

Mathematics Subject Classification

65L05 

Notes

Acknowledgements

The results reported in this paper constitute part of the research carried out during the visit of the second author to the University of Tabriz which was supported by the Ministry of Science, Research and Technology of Iran. This author wishes to express her gratitude to A. Abdi for making this visit possible.

References

  1. 1.
    Abdi, A.: Construction of high-order quadratically stable second-derivative general linear methods for the numerical integration of stiff ODEs. J. Comput. Appl. Math. 303, 218–228 (2016)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Abdi, A., Braś, M., Hojjati, G.: On the construction of second derivative diagonally implicit multistage integration methods. Appl. Numer. Math. 76, 1–18 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Abdi, A., Butcher, J.C.: Order bounds for second derivative approximations. BIT Numer. Math. 52, 273–281 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Abdi, A., Hojjati, G.: An extension of general linear methods. Numer. Algor. 57, 149–167 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Abdi, A., Hojjati, G.: Implementation of Nordsieck second derivative methods for stiff ODEs. Appl. Numer. Math. 94, 241–253 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Abdi, A., Hojjati, G.: Maximal order for second derivative general linear methods with Runge–Kutta stability. Appl. Numer. Math. 61, 1046–1058 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Braś, M.: Nordsieck methods with inherent quadratic stability. Math. Model. Anal. 16, 82–96 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Butcher, J.C.: General linear methods. Acta Numer. 15, 157–256 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Butcher, J.C.: General linear methods for stiff differential equations. BIT Numer. Math. 41, 240–264 (2001)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, New York (2016)CrossRefMATHGoogle Scholar
  11. 11.
    Butcher, J.C.: On the convergence of numerical solutions to ordinary differential equations. Math. Comput. 20, 1–10 (1996)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Butcher, J.C., Hojjati, G.: Second derivative methods with RK stability. Numer. Algor. 40, 415–429 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Butcher, J.C., Jackiewicz, Z.: Construction of diagonally implicit general linear methods type 1 and 2 for ordinary differential equations. Appl. Numer. Math. 21, 385–415 (1996)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Butcher, J.C., Jackiewicz, Z.: Diagonally implicit general linear methods for ordinary differential equations. BIT Numer. Math. 33, 452–472 (1993)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Butcher, J.C., Jackiewicz, Z.: Implementation of diagonally implicit multistage integration methods for ordinary differential equations. SIAM J. Numer. Anal. 34, 2119–2141 (1997)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Butcher, J.C., Wright, W.: A transformation relating explicit and diagonally-implicit general linear methods. Appl. Numer. Math. 44, 313–327 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Butcher, J.C., Wright, W.: The construction of practical general linear methods. BIT Numer. Math. 43, 695–721 (2003)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Cardone, A., Jackiewicz, Z.: Explicit Nordsieck methods with quadratic stability. Numer. Algor. 60, 1–25 (2012)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Cash, J.R.: Second derivative extended backward differentiation formulas for the numerical integration of stiff systems. SIAM J. Numer. Anal. 18, 21–36 (1981)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Chan, R.P.K., Tsai, A.Y.J.: On explicit two-derivative Runge–Kutta methods. Numer. Algor. 53, 171–194 (2010)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Conte, D., D’Ambrosio, R., Jackiewicz, Z.: Two-step Runge–Kutta methods with quadratic stability functions. J. Sci. Comput. 44, 191–218 (2010)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Enright, W.H.: Second derivative multistep methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 11, 321–331 (1974)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Ezzeddine, A.K., Hojjati, G., Abdi, A.: Sequential second derivative general linear methods for stiff systems. Bull. Iran. Math. Soc. 40, 83–100 (2014)MathSciNetMATHGoogle Scholar
  24. 24.
    Ezzeddine, A.K., Hojjati, G., Abdi, A.: Perturbed second derivative multistep methods. J. Numer. Math. 23, 235–245 (2015)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, Berlin (1996)CrossRefMATHGoogle Scholar
  26. 26.
    Hojjati, G., Rahimi Ardabili, M.Y., Hosseini, S.M.: New second derivative multistep methods for stiff systems. Appl. Math. Model. 30, 466–476 (2006)CrossRefMATHGoogle Scholar
  27. 27.
    Hosseini Nasab, M., Hojjati, G., Abdi, A.: G-symplectic second derivative general linear methods for Hamiltonian problems. J. Comput. Appl. Math. 313, 486–498 (2017)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Hosseini Nasab, M., Abdi, A., Hojjati, G.: Symmetric second derivative integration methods. J. Comput. Appl. Math. 330, 618–629 (2018)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Huang, S.J.Y.: Implementation of general linear methods for stiff ordinary differential equations. Ph.D. thesis, Department of mathematics, The University of Auckland (2005)Google Scholar
  30. 30.
    Jackiewicz, Z.: General Linear Methods for Ordinary Differential Equations. Wiley, New Jersey (2009)CrossRefMATHGoogle Scholar
  31. 31.
    Lambert, J.D.: Computational Methods in Ordinary Differential Equations. Wiley, New York (1973)MATHGoogle Scholar
  32. 32.
    Lee, J.H.J.: Numerical methods for ordinary differential equations: a survey of some standard methods. M.Sc. thesis, Department of mathematics, The University of Auckland (2004)Google Scholar
  33. 33.
    Movahedinejad, A., Hojjati, G., Abdi, A.: Second derivative general linear methods with inherent Runge–Kutta stability. Numer. Algor. 73, 371–389 (2016)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Movahedinejad, A., Hojjati, G., Abdi, A.: Construction of Nordsieck second derivative general linear methods with inherent quadratic stability. Math. Model. Anal. 22, 60–77 (2017)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Shampine, L.F., Reichelt, M.W.: The MATLAB ODE suite. SIAM J. Sci. Comput. 18, 1–22 (1997)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Wright, W.M.: Expilicit general linear methods with inherent Runge–Kutta stability. Numer. Algor. 31, 381–399 (2002)CrossRefMATHGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica del Consiglio Nazionale delle Ricerche 2018

Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesUniversity of TabrizTabrizIran
  2. 2.Department of MathematicsLorestan UniversityKhorramabadIran

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