Advertisement

Computational and Applied Mathematics

, Volume 37, Issue 5, pp 6920–6954 | Cite as

On explicit two-derivative two-step Runge–Kutta methods

  • Mukaddes Ökten Turaci Email author
  • Turgut Öziş
Article
  • 302 Downloads

Abstract

We introduce a class of methods for the numerical solution of ordinary differential equations. These methods called as two-derivative two-step Runge–Kutta methods are extension of the two-step Runge–Kutta methods in which the second derivative of the solution is included. These methods are a special class of second-derivative general linear methods studied by many authors Butcher et al. (Numer Algorithms 40:415–429, 2005), Abdi et al. (Numer Algorithms 57:149–167, 2011), Okuonghae and Ikhile (Numer Algorithms 67(3):637–654, 2014). The order conditions are derived based on the algebraic theory of Butcher (Mathe Comput 26:79–106, 1972) and the \(\mathcal {B}-\)series theory Hairer and Wanner (Computing 13:1–15, 1974), in a similar way to Chan and Chan (2006). In this study, special explicit two-derivative two-step Runge–Kutta methods that possess one evaluation of the first derivative and many evaluations of the second derivative per step are introduced. Methods with stages up to five and of order up to eight are presented. The numerical calculations have been performed on some non-stiff and mildly stiff problems and comparisons have been made with the accessible methods in the literature.

Keywords

Two-step Runge–Kutta methods Second-derivative general linear methods Two-derivative Runge–Kutta methods Rooted trees Order conditions 

Mathematics Subject Classification

65L05 65L06 65L20 

Notes

Acknowledgements

The authors would like to thank the referees and the editor for their valuable comments and suggestions which improved the presentation of the paper.

References

  1. Abdi A, Hojjati G (2011a) An extension of general linear methods. Numerical Algorithms 57:149–167Google Scholar
  2. Abdi A, Hojjati G (2011b) Maximal order for second derivative general linear methods with Runge-Kutta stability. Applied Numerical Mathematics 61:1046–1058Google Scholar
  3. Abdi A, Hojjati G (2015) Implementation of nordsieck second derivative methods for stiff odes. Applied Numerical Mathematics 94:241–253MathSciNetzbMATHCrossRefGoogle Scholar
  4. Akanbi MA, Okunuga SA, Sofoluwe AB (2012) Step size bounds for a class of multiderivative explicit Runge-Kutta methods, Modeling and Simulation in Engineering Economics and Management 188–197, Springer, BerlinGoogle Scholar
  5. Albrecht P (1985) Numerical treatment of o.d.e.s: the theory of a-methods. Numerische Mathematik 47:59–87MathSciNetzbMATHCrossRefGoogle Scholar
  6. Albrecht P (1987) A new theoretical approach to Runge-Kutta methods. SIAM Journal on Numerical Analysis 24(2):391–406MathSciNetzbMATHCrossRefGoogle Scholar
  7. Bartoszewski Z, Jackiewicz Z (1998) Construction of two-step Runge-Kutta methods of high order for ordinary differential equations. Numerical Algorithms 18:51–70MathSciNetzbMATHCrossRefGoogle Scholar
  8. Burrage K (1988) Order properties of implicit multivalue methods for ordinary differential equations. IMA Journal of Numerical Analysis 8:43–69MathSciNetzbMATHCrossRefGoogle Scholar
  9. Butcher JC (1972) An algebraic theory of integration methods. Mathematics of Computation 26:79–106MathSciNetzbMATHCrossRefGoogle Scholar
  10. Butcher JC (1987) The Numerical Analysis of Ordinary Differential Equations. :Runge-Kutta and General Linear Methods. Wiley, New YorkzbMATHGoogle Scholar
  11. Butcher JC (2008) Numerical Methods for Ordinary Differential Equations. John Wiley & Sons, ChichesterzbMATHCrossRefGoogle Scholar
  12. Butcher JC (2010) Trees and numerical methods for ordinary differential equations. Numerical Algorithms 53:153–170MathSciNetzbMATHCrossRefGoogle Scholar
  13. Butcher JC, Hojjati G (2005) Second derivative methods with RK stability. Numerical Algorithms 40:415–429MathSciNetzbMATHCrossRefGoogle Scholar
  14. Butcher JC, Tracogna S (1997) Order conditions for two-step Runge-Kutta methods. Applied Numerical Mathematics 24:351–364MathSciNetzbMATHCrossRefGoogle Scholar
  15. Byrne GD, Lambert RJ (1966) Pseudo Runge-Kutta methods involving two points. Journal of the ACM (JACM) 13:114–123MathSciNetzbMATHCrossRefGoogle Scholar
  16. Cash JR (1981) Second derivative extended backward differentiation formulas for the numerical integration of stiff systems. SIAM Journal on Numerical Analysis 18(1):21–36MathSciNetzbMATHCrossRefGoogle Scholar
  17. Chan TMH, Chan RPK (2006) A simplified approach to the order conditions of integration methods. Computing 77(3):237–252MathSciNetzbMATHCrossRefGoogle Scholar
  18. Chan RPK, Tsai AYJ (2010) On explicit two-derivative Runge-Kutta methods. Numerical Algorithms 53:171–194MathSciNetzbMATHCrossRefGoogle Scholar
  19. Chollom J, Jackiewicz Z (2003) Construction of two-step Runge-Kutta methods with large regions of absolute stability. Journal of Computational and Applied Mathematics 157:125–137MathSciNetzbMATHCrossRefGoogle Scholar
  20. D’Ambrosio R, Esposito E, Paternoster B (2012) Exponentially fitted two-step Runge-Kutta methods: construction and parameter selection. Applied Mathematics and Computation 218(14):7468–7480MathSciNetzbMATHCrossRefGoogle Scholar
  21. Enright WH (1974) Second derivative multistep methods for stiff ordinary differential equations. SIAM Journal on Numerical Analysis 11(2):321–331MathSciNetzbMATHCrossRefGoogle Scholar
  22. Ezzeddine AK, Hojjati G (2012) Third derivative multistep methods for stiff systems. International Journal of Nonlinear Science 14(4):443–450MathSciNetzbMATHGoogle Scholar
  23. Gear CW (1971) Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, NJzbMATHGoogle Scholar
  24. Gekeler E, Widmann R (1986) On the order conditions of Runge-Kutta methods with higher derivatives. Numerische Mathematik 50(2):183–203MathSciNetzbMATHCrossRefGoogle Scholar
  25. Goeken D, Johnson O (1999) Fifth-order Runge-Kutta with higher order derivative approximations. Electronic Journal of Differential Equations Conference 02:1–9MathSciNetzbMATHGoogle Scholar
  26. Goeken D, Johnson O (2000) Runge-Kutta with higher order derivative approximations. Applied Numerical Mathematics 34(2–3):207–218MathSciNetzbMATHCrossRefGoogle Scholar
  27. Hairer E, Wanner G (1973) Multistep-multistage-multiderivative methods for ordinary differential equations. Computing 11(3):287–303MathSciNetzbMATHCrossRefGoogle Scholar
  28. Hairer E, Wanner G (1974) On the butcher group and general multi-value methods. Computing 13:1–15MathSciNetzbMATHCrossRefGoogle Scholar
  29. Hairer E, Wanner G (1996) Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, 2nd edn., Springer Series in Computational Mathematics, vol. 14. Springer, BerlinGoogle Scholar
  30. Hairer E, Wanner G (1997) Order conditions for general two-step Runge-Kutta methods. SIAM Journal on Numerical Analysis 34(6):2087–2089MathSciNetzbMATHCrossRefGoogle Scholar
  31. Hairer E, Nørsett SP, Wanner G (1993) Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd edn., Springer Series in Computational Mathematics, vol. 8. Springer, BerlinGoogle Scholar
  32. Hojjati G, Rahimi Ardabili MY, Hosseini SM (2006) New second derivative multistep methods for stiff systems. Applied Mathematical Modelling 30(5):466–476zbMATHCrossRefGoogle Scholar
  33. Jackiewicz Z (2009) General Linear Methods for Ordinary Differential Equations. John Wiley & Sons, Hoboken, New JerseyzbMATHCrossRefGoogle Scholar
  34. Jackiewicz Z, Tracogna S (1995) A general class of two-step Runge-Kutta methods for ordinary differential equations. SIAM Journal on Numerical Analysis 32(5):1390–1427MathSciNetzbMATHCrossRefGoogle Scholar
  35. Jackiewicz Z, Verner JH (2002) Derivation and implementation of two-step Runge-Kutta pairs. Japan Journal of Industrial and Applied Mathematics 19:227–248MathSciNetzbMATHCrossRefGoogle Scholar
  36. Jackiewicz Z, Renaut R, Feldstein A (1991) Two-step Runge-Kutta methods. SIAM Journal on Numerical Analysis 28(4):1165–1182MathSciNetzbMATHCrossRefGoogle Scholar
  37. Jackiewicz Z, Renaut RA, Zennaro M (1995) Explicit two-step Runge-Kutta methods. Applications of Mathematics 40(6):433–456MathSciNetzbMATHGoogle Scholar
  38. Lambert JD (1973) Computational Methods in Ordinary Differential Equations. Wiley, New YorkzbMATHGoogle Scholar
  39. Ökten Turacı M, Öziş T (2017) Derivation of three-derivative Runge-Kutta methods. Numerical Algorithms 74(1):247–265MathSciNetzbMATHCrossRefGoogle Scholar
  40. Okuonghae RI, Ikhile MNO (2014) Second derivative general linear methods. Numerical Algorithms 67(3):637–654MathSciNetzbMATHCrossRefGoogle Scholar
  41. Phohomsiri P, Udwadia FE (2004) Acceleration of Runge-Kutta integration schemes. Discrete Dynamics in Nature and Society 2004(2):307–314MathSciNetzbMATHCrossRefGoogle Scholar
  42. Rabiei F, Ismail F (2011) Third-order improved Runge-Kutta method for solving ordinary differential equation. International Journal of Applied Physics and Mathematics 1(3):191–194CrossRefGoogle Scholar
  43. Reynoso Gustavo Franco, Gottlieb Sigal,Grant Zachary J (2017) Strong stability preserving sixth order two-derivative Runge–Kutta methods, in: AIP Conference Proceedings, Vol. 1863, AIP Publishing, p. 560068Google Scholar
  44. Rosenbrock HH (1963) Some general implicit processes for the numerical solution of differential equations. The Computer Journal 5:329–330MathSciNetzbMATHCrossRefGoogle Scholar
  45. Seal DC, Güçlü Y, Christlieb AJ (2014) High-order multiderivative time integrators for hyperbolic conservation laws. Journal of Scientific Computing 60:101–140MathSciNetzbMATHCrossRefGoogle Scholar
  46. Skvortsov LM (2010) Explicit two-step Runge-Kutta methods. Mathematical Models and Computer Simulations 2(2):222–231MathSciNetCrossRefGoogle Scholar
  47. Süli E (2014) Numerical Solution of Ordinary Differential Equations. University of Oxford,Google Scholar
  48. Tsai AYJ, Chan RPK, Wang S (2014) Two-derivative Runge-Kutta methods for PDEs using a novel discretization approach. Numerical Algorithms 65:687–703MathSciNetzbMATHCrossRefGoogle Scholar
  49. Udwadia FE, Farahani A (2008) Accelerated Runge-Kutta methods, Discrete Dynamics in Nature and Society 2008, 38 pages, Article ID 790619Google Scholar
  50. Verner JH (1978) Explicit Runge-Kutta methods with estimates of the local truncation error. SIAM Journal on Numerical Analysis 15(4):772–790MathSciNetzbMATHCrossRefGoogle Scholar
  51. Watt JM (1967) The asymptotic discretization error of a class of methods for solving ordinary differential equations. Proc. Cambridge Philos. Soc. 63(2):461–472MathSciNetzbMATHCrossRefGoogle Scholar
  52. Wu X (2003) A class of Runge-Kutta formulae of order three and four with reduced evaluations of function. Applied Mathematics and Computation 146(2–3):417–432MathSciNetzbMATHCrossRefGoogle Scholar
  53. Wu X, Xia J (2006) Extended Runge-Kutta-like formulae. Applied Numerical Mathematics 56:1584–1605MathSciNetzbMATHCrossRefGoogle Scholar
  54. Wusu AS, Akanbi MA, Okunuga SA (2013) A three-stage multiderivative explicit Runge-Kutta method. American Journal of Computational Mathematics 3:121–126CrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018

Authors and Affiliations

  1. 1.Department of Computer Programming, Yenice Vocational SchoolKarabük UniversityKarabükTurkey
  2. 2.Department of MathematicsEge UniversityİzmirTurkey

Personalised recommendations