ORDER OF THE RUNGE-KUTTA METHOD AND EVOLUTION OF THE STABILITY REGION

Hippolyte Séka     (Institut National Polytechnique Houphouet-Boigny, Côte d'Ivoire)
Kouassi Richard Assui     (Institut National Polytechnique Houphouët–Boigny, BP 1093 Yamoussoukro, Côte d'Ivoire)

Abstract


In this article, we demonstrate through specific examples that the evolution of the size of the absolute stability regions of Runge–Kutta methods for ordinary differential equation does not depend on the order of methods.


Keywords


Stability region, Runge–Kutta methods, Ordinary differential equations, Order of methods.

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References


  1. Butcher J-C. Numerical Methods for Ordinary Differential Equations. 2nd ed. John Wiley & Sons Ltd., 2008. 175 p. DOI: 10.1002/9780470753767
  2. Calvo M., Montijano J.I., Randez L. A new embedded pair of Runge–Kutta formulas of orders 5 and 6. Comput. Math. Appl., 1990. Vol. 20, No. 1. P. 15–24. DOI: 10.1016/0898-1221(90)90064-Q
  3. Cassity C.R. The complete solution of the fifth order Runge–Kutta equations. SIAM J. Numer. Anal., 1969. Vol. 6, No. 3. P. 432–436. DOI: 10.1137/0706038
  4. Feagin T. A tenth-order Runge–Kutta method with error estimate. In: Proc. of the IAENG Conf. on Scientific Computing. Hong Kong, 2007. Accessible at https://sce.uhcl.edu/feagin/courses/rk10.pdf
  5. Feagin T. High-Order Explicit Runge-Kutta Methods. 2013. Accessible at http://sce.uhcl.edu/rungekutta
  6. Hairer E., Nørsett S.P., Wanner G. Solving Ordinary Differential Equations I. Nonstiff Problems. Springer Ser. Comput. Math., vol. 8. Berlin, Heidelberg: Springer–Verlag, 1993. 528 p. DOI: 10.1007/978-3-540-78862-1
  7. Houben S. Stability Regions of Runge–Kutta Methods. Eindhoven University of Technology, 2002. Accessible at URL: https://www.win.tue.nl/casa/meetings/seminar/previous/_abstract020220_files/talk.pd
  8. Jackiewicz Z. General Linear Methods for Ordinary Differential Equations. John Wiley & Sons, Inc., 2009. 482 p. DOI: 10.1002/9780470522165
  9. Khashin SI. List of Some Known Runge–Kutta Methods Family. Preliminary version. 2013. Accessible at URL: http://math.ivanovo.ac.ru/dalgebra/Khashin/rk/sh_rk.html
  10. Kashin S.I. Estimating the error in classical Runge–Kutta methods. Comput. Math. Math. Phys., 2014. Vol. 54, No. 5. P. 767–774. DOI: 10.1134/S0965542514050145
  11. Liu M.Z., Song M.H., Yang Z.W. Stability of Runge–Kutta methods in the numerical solution of equation \(u'(t)=au(t)+a_{0}u([t])\). J. Comput. Appl. Math., 2004. Vol. 166, No. 2. P. 361–370. DOI: 10.1016/j.cam.2003.04.002
  12. Seka H., Assui K.R. A New Eighth Order Runge-Kutta Family Method. J. Math. Res., 2019. Vol. 11, No. 2. P. 190–199. DOI: 10.5539/jmr.v11n2p190
  13. Velagala S.R. Stability Analysis of the 4th order Runge–Kutta Method in Application to Colloidal Particle Interactions. Master’s thesis. University of Illinois, Urbana-Champaign, USA, 2014. Accessible at http://hdl.handle.net/2142/72750



DOI: http://dx.doi.org/10.15826/umj.2019.2.006

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