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New class of hybrid BDF methods for the computation of numerical solutions of IVPs

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Abstract

A new class of hybrid BDF-like methods is presented for solving systems of ordinary differential equations (ODEs) by using the second derivative of the solution in the stage equation of class 2 + 1hybrid BDF-like methods to improve the order and stability regions of these methods. An off-step point, together with two step points, has been used in the first derivative of the solution, and the stability domains of the new methods have been obtained by showing that these methods are A-stable for order p, p = 3,4,5,6,7and A(α)-stable for order p, 8 ≤ p ≤ 14. The numerical results are also given for four test problems by using variable and fixed step-size implementations.

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References

  1. Ebadi, M., Gokhale, M. Y.: Class 2 + 1 hybrid BDF-like methods for the numerical solutions of ordinary differential equations. Calcolo 48(4), 273–291 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ebadi, M., Gokhale, M. Y.: Hybrid BDF methods for the numerical solutions of ordinary differential equations. Numer. Algorithms 55, 1–17 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Burrage, K., Butcher, J. C., Chipman, F. H.: An implementation of singly-implicit Runge–Kutta methods. BIT 20, 326–340 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burrage, K.: A special family of Runge–Kutta methods for solving stiff differential equations. BIT 18, 22–41 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Butcher, J. C., Wright, W. M.: Applications of doubly companion matrices. Appl. Numer. Math. 56, 358–373 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alexander, R.: Diagonally implicit Runge-Kutta methods for stiff ODEs. SIAM J. Nume. Anal. 14, 1006–10021 (1977)

    Article  MATH  Google Scholar 

  7. Butcher, J. C., Diamantakis, M.: DESIRE: diagonally extended singly-implicit Runge-Kutta effective order methods. Numer. Algorithms 17, 121–145 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Butcher, J. C., Cash, J. R., Diamantakis, M.: DESI Methods for stiff initial value problems. ACM Trans. Math. Software 22, 401–422 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kelley, C. T.: Solving nonlinear equations with Newton’s method, Fundamentals of Algorithms. SIAM, Philadelphia (2003)

    Book  Google Scholar 

  10. Lambert, J. D.: Computational methods in ordinary differential equations, pp 143–144. Wiley, London (1972)

    Google Scholar 

  11. Hairer, E., Wanner, G.: Solving ordinary differential equations II: stiff and differential algebraic problem. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

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Acknowledgements

The author is very grateful to the referees for their useful and valuable comments.

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Correspondence to Moosa Ebadi.

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Ebadi, M. New class of hybrid BDF methods for the computation of numerical solutions of IVPs. Numer Algor 79, 179–193 (2018). https://doi.org/10.1007/s11075-017-0433-7

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  • DOI: https://doi.org/10.1007/s11075-017-0433-7

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