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Runge–Kutta Methods for Ordinary Differential Equations

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Numerical Analysis and Optimization

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 134))

Abstract

Since their first discovery by Runge (Math Ann 46:167–178, 1895), Heun (Z Math Phys 45:23–38, 1900) and Kutta (Z Math Phys 46:435–453, 1901), Runge–Kutta methods have been one of the most important procedures for the numerical solution of ordinary differential equation systems. This survey paper ranges over many aspects of Runge–Kutta methods, including order conditions, order barriers, the efficient implementation of implicit methods, effective order methods and strong stability-preserving methods. Finally, applications to the analysis and implementation of G-symplectic methods will be discussed.

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References

  1. Alexander, R.: Diagonally implicit Runge–Kutta methods for stiff ODEs. SIAM J. Numer. Anal. 14, 1006–1021 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  2. Butcher, J.C.: Coefficients for the study of Runge–Kutta integration processes. J. Austral. Math. Soc. 3, 185–201 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  3. Butcher, J.C.: On the attainable order of Runge–Kutta methods. Math. Comput. 19, 408–417 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  4. Butcher, J.C.: An algebraic theory of integration methods. Math. Comput. 26, 79–106 (1972)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  6. Butcher, J.C.: Dealing with parasitic behaviour in G-symplectic integrators. In: Ansorge, R. (ed.) Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, pp. 105–123. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  7. Butcher, J.C.: The cohesiveness of G-symplectic methods. Numer. Algorithms (2015) DOI:10.1007/s11075-015-9964-y

    Google Scholar 

  8. Butcher, J.C., Cash, J.R.: Towards efficient Runge–Kutta methods for stiff systems. SIAM J. Numer. Anal. 27, 753–761 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  9. Butcher, J.C., Chartier, P.: A generalization of singly-implicit Runge–Kutta methods. Appl. Numer. Math. 24, 343–350 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Butcher, J.C., Habib, Y., Hill, A.T., Norton, T.J.T.: The control of parasitism in G-symplectic methods. SIAM J. Numer. Anal. 52, 2440–2465 (2014)

    Article  MathSciNet  Google Scholar 

  11. Gill, S.: A process for the step-by-step integration of differential equations in an automatic computing machine. Proc. Camb. Philos. Soc. 47, 96–108 (1951)

    Article  MATH  Google Scholar 

  12. Gottlieb, S., Ketcheson, D.I., Shu, C.-W.: Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations. World Scientific Press, Singapore (2011)

    Book  MATH  Google Scholar 

  13. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 1st edn. Springer, New York (2003)

    Google Scholar 

  14. Heun, K.: Neue Methoden zur approximativen Integration der Differentialgleichungen einer unabhängigen Veränderlichen. Z. Math. Phys. 45, 23–38 (1900)

    MATH  Google Scholar 

  15. Kutta, W.: Beitrag zur näherungsweisen Integration totaler Differentialgleichungen. Z. Math. Phys. 46, 435–453 (1901)

    MATH  Google Scholar 

  16. Merson, R.H.: An operational method for the study of integration processes. In: Proceedings of Symposium on Data Processing. Weapons Research Establishment, Salisbury, S. Australia (1957)

    Google Scholar 

  17. Nyström, E.J.: Über die numerische integration von differentialgleichungen. Acta Soc. Sci. Fennicae 50(13), 55pp. (1925)

    Google Scholar 

  18. Runge, C.: Über die numerische auflösung von differentialgleichungen. Math. Ann. 46, 167–178 (1895)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sanz-Serna, J.M.: Runge–Kutta schemes for Hamiltonian systems. BIT 28, 877–883 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shu, C.-W.: Total-variation diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9, 1073–1084 (1988)

    Article  MATH  Google Scholar 

  21. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shockcapturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author expresses his thanks for support from the Marsden Fund and for helpful comments from an anonymous referee.

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Authors and Affiliations

  1. The University of Auckland, Auckland, New Zealand

    J. C. Butcher

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  1. J. C. Butcher
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Correspondence to J. C. Butcher .

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Editors and Affiliations

  1. Department of Mathematics College of Science, Sultan Qaboos University, Muscat, Oman

    Mehiddin Al-Baali

  2. Faculty of Engineering, University of Calabria, Arcavacada di Rende, Italy

    Lucio Grandinetti

  3. Department Mathematics and Statistics College of Science, Sultan Qaboos University, Muscat, Oman

    Anton Purnama

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Butcher, J.C. (2015). Runge–Kutta Methods for Ordinary Differential Equations. In: Al-Baali, M., Grandinetti, L., Purnama, A. (eds) Numerical Analysis and Optimization. Springer Proceedings in Mathematics & Statistics, vol 134. Springer, Cham. https://doi.org/10.1007/978-3-319-17689-5_2

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