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Explicit adaptive Runge-Kutta methods for stiff and oscillation problems

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Abstract

Explicit Runge-Kutta methods with the coefficients tuned to the problem of interest are examined. The tuning is based on estimates for the dominant eigenvalues of the Jacobian matrix obtained from the results of the preliminary stages. Test examples demonstrate that methods of this type can be efficient in solving stiff and oscillation problems.

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References

  1. L. M. Skvortsov, “Explicit Multistep Method for the Numerical Solution of Stiff Differential Equations,” Comput. Math. Math. Phys. 47, 915–923 (2007).

    Article  MathSciNet  Google Scholar 

  2. M. E. Fowler and R. M. Warten, “A Numerical Integration Technique for Ordinary Differential Equations with Widely Separated Eigenvalues,” IBM J. Research Development 11(5), 537–543 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  3. S. O. Fatunla, “Numerical Integrators for Stiff and Highly Oscillatory Differential Equations,” Math. Comput. 34(150), 373–390 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  4. V. V. Bobkov, “A Technique for the Construction of Methods for the Numerical Solution of Differential Equations,” Differ. Uravn. Ikh Primen. 19(7), 1115–1122 (1983).

    MathSciNet  MATH  Google Scholar 

  5. A. N. Zavorin, “Application of Nonlinear Methods for the Calculation of Transient Processes in Electric Circuits,” Izv. Vyssh. Uchebn. Zaved., Radioelektronika 26(3), 35–41 (1983).

    Google Scholar 

  6. A. N. Saworin, “Über die Effektivitat Einiger Nichtlinearer Verfahren bei der Numerischen Behandlung Steifer Differentialgleichungssysteme,” Numer. Math. 40, 169–177 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  7. S. S. Ashour and O. T. Hanna, “Explicit Exponential Method for the Integration of Stiff Ordinary Differential Equations,” J. of Guidance, Control and Dynamics 14(6), 1234–1239 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  8. J. D. Lambert, “Nonlinear Methods for Stiff Systems of Ordinary Differential Equations,” Lect. Notes Math. 363, 75–88 (1974).

    Article  Google Scholar 

  9. A. Wambecq, “Rational Runge-Kutta Methods for Solving Systems of Ordinary Differential Equations,” Computing 20(4), 333–342 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  10. V. V. Bobkov, “New Explicit A-Stable Methods for the Numerical Solution of Differential Equations,” Differ. Uravn. 14(12), 2249–2251 (1978).

    MathSciNet  MATH  Google Scholar 

  11. X. Y. Wu and J. L. Xia, “Two Low Accuracy Methods for Stiff Systems,” Appl. Math. Comput. 123(2), 141–153 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  12. H. Ramos, “A Non-Standard Explicit Integration Scheme for Initial-Value Problems,” Appl. Math. Comput. 189(1), 710–718 (2007).

    Article  MathSciNet  Google Scholar 

  13. L. M. Skvortsov, “Adaptive Methods for the Digital Modeling of Dynamical Systems,” Izv. Ross. Akad. Nauk, Teor. Sist. Upr., No. 4, 180–190 (1995).

  14. L. M. Skvortsov, “Adaptivnye metody chislennogo integrirovaniya v zadachakh modelirovaniya dinamicheskikh sistem,” J. Comput. Syst. Sci. Int. 38, 573–579 (1999).

    MATH  Google Scholar 

  15. L. M. Skvortsov, “Explicit Adaptive Methods for the Numerical Solution of Stiff Systems,” Matem. Modelir. 12(12), 97–107 (2000).

    MathSciNet  MATH  Google Scholar 

  16. L. M. Skvortsov, “Explicit Runge-Kutta Methods for Moderately Stiff Problems,” Comput. Math. Math. Phys. 45, 1939–1951 (2005).

    MathSciNet  Google Scholar 

  17. V. I. Lebedev, “How to Solve Stiff Systems of Differential Equations Using Explicit Methods,” in Vychislitel’nye protsessy i sistemy (Nauka, Moscow, 1991), No. 8, pp. 237–291 [in Russian].

    Google Scholar 

  18. T. E. Simos, “A Modified Runge-Kutta Method for the Numerical Solution of ODE’s with Oscillation Solutions,” Appl. Math. Lett. 9(6), 61–66 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  19. J. M. Franco, “Runge-Kutta Methods Adapted to the Numerical Integration of Oscillatory Problems,” Appl. Numer. Math. 50, 427–443 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  20. Y. Fang, Y. Song, and X. Wu, “New Embedded Pairs of Explicit Runge-Kutta Methods with FSAL Properties Adapted to the Numerical Integration of Oscillatory Problems,” Phys. Lett. A 372, 6551–6559 (2008).

    Article  MathSciNet  Google Scholar 

  21. A. A. Kosti, Z. A. Anastassi, and T. E. Simos, “An Optimized Explicit Runge-Kutta Method with Increased Phase-Lag Order for the Numerical Solution of the Schrodinger Equation and Related Problems,” J. Math. Chem. 47(1), 315–330 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Prothero and A. Robinson, “On the Stability and Accuracy of One-Step Methods for Solving Stiff Systems of Ordinary Differential Equations,” Math. Comput. 28(1), 145–162 (1974).

    Article  MathSciNet  Google Scholar 

  23. E. Hairer and G. Wanner, Solving Ordinary Differential Equations, Vol. 2: Stiff and Differential-Algebraic Problems (Springer, Berlin, 1987–1991; Mir, Moscow, 1999).

    Google Scholar 

  24. F. Mazzia and C. Magherini, “Test Set for Initial Value Problem Solvers,” Release, 4 (2008); http://pitagora.dm.uniba.it/~testset/reprt/testset.pdf.

  25. http://web.math.unifi.it/users/brugnano/BiM/BiMD/index-BiMD.htm.

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Authors and Affiliations

  1. Bauman State Technical University, Vtoraya Baumanskaya ul. 5, Moscow, 105005, Russia

    L. M. Skvortsov

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  1. L. M. Skvortsov
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Correspondence to L. M. Skvortsov.

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Original Russian Text © L.M. Skvortsov, 2011, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2011, Vol. 51, No. 8, pp. 1434–1448.

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Skvortsov, L.M. Explicit adaptive Runge-Kutta methods for stiff and oscillation problems. Comput. Math. and Math. Phys. 51, 1339–1352 (2011). https://doi.org/10.1134/S0965542511080173

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