Kyungpook Mathematical Journal

  • 제56권2호
  • /
  • Pages.311-327
  • /
  • 2016
  • /
  • 1225-6951(pISSN)
  • /
  • 0454-8124(eISSN)
경북대학교 자연과학대학 수학과 (Department of Mathematics, Kyungpook National University)
권호 표지
DOI QR코드

DOI QR Code

An Error Embedded Runge-Kutta Method for Initial Value Problems

  • Bu, Sunyoung (Department of liberal arts, Hongik University) ;
  • Jung, WonKyu (Research & Technology Division, Dongwoo Fine-Chem) ;
  • Kim, Philsu (Department of Mathematics, Kyungpook National University)
  • 투고 : 2015.06.28
  • 심사 : 2015.10.20
  • 발행 : 2016.06.23
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we propose an error embedded Runge-Kutta method to improve the traditional embedded Runge-Kutta method. The proposed scheme can be applied into most explicit embedded Runge-Kutta methods. At each integration step, the proposed method is comprised of two equations for the solution and the error, respectively. These solution and error are obtained by solving an initial value problem whose solution has the information of the error at each integration step. The constructed algorithm controls both the error and the time step size simultaneously and possesses a good performance in the computational cost compared to the original method. For the assessment of the effectiveness, the van der Pol equation and another one having a difficulty for the global error control are numerically solved. Finally, a two-body Kepler problem is also used to assess the efficiency of the proposed algorithm.

키워드

참고문헌

  1. https://www.dm.uniba.it/testset/testsetivpsolvers
  2. K. E. Atkinson, An introduction to numerical analysis, John Wiley & Sons, Inc., 1989.
  3. L. Brugnano, F. Iavernaro and D. Trigiante, Energy- and quadratic invariantspreserving integrators based upon Gauss collocation formulae, SIAM J. Numer. Anal., 50(2012), 2897-2916. https://doi.org/10.1137/110856617
  4. S. Bu, J. Huang and M. L. Minion, Semi-implicit Krylov deferred correction methods for differential algebraic equations, Math. Comput., 81(280)(2012), 2127-2157. https://doi.org/10.1090/S0025-5718-2012-02564-6
  5. M. P. Calvo and E. Hairer, Accurate long-term integration of dynamical systems, Appl. Numer. Math., 18(1995), 95-105. https://doi.org/10.1016/0168-9274(95)00046-W
  6. J. R. Dormand and P. J. Prince, High order embedded Runge-Kutta formulae, J. Comput. Appl. Math., 7(1)(1981), 67-75. https://doi.org/10.1016/0771-050X(81)90010-3
  7. E. Fehlberg, Classical fifth-, sixth-, seventh-, and eighth-order Runge-Kutta formulas with stepsize control, NASA; for sale by the Clearinghouse for Federal Scientific and Technical Information, Springfield, VA, 1968.
  8. C. W. Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, 1971.
  9. K. Gustafsson, Control-theoretic techniques for stepsize selection in implicit Runge-Kutta methods, ACM Trans. Math. Softw., 20(4)(1994), 496-517. https://doi.org/10.1145/198429.198437
  10. E. Hairer, Long-time integration of non-stiff and oscillatory Hamiltonian systems, AIP Conf. Proc., 1168(1)(2009), 3-6.
  11. E. Hairer, S. P. Norsett and G. Wanner, Solving ordinary differential equations. I nonstiff, Springer Series in Computational Mathematics, Springer, 1993.
  12. E. Hairer and G. Wanner, Solving ordinary differential equations. II stiff and differential-algebraic problems, Springer Series in Computational Mathematics, Springer, 1996.
  13. C. Johnson, Error estimates and adaptive time-step control for a class of one-step methods for stiff ordinary differential equations, SIAM J. Numer. Anal., 25(4)(1988), 908-926. https://doi.org/10.1137/0725051
  14. D. Kavetski, P. Binning and S. W. Sloan, Adaptive time stepping and error control in a mass conservative numerical solution of the mixed form of Richards equation, Adv. Water Resour., 24(2001), 595-605. https://doi.org/10.1016/S0309-1708(00)00076-2
  15. P. Kim, X. Piao and S. D. Kim, An error corrected Euler method for solving stiff problems based on Chebyshev collocation, SIAM J. Numer. Anal. 49(2011), 2211-2230. https://doi.org/10.1137/100808691
  16. S. D. Kim, X. Piao, D. H. Kim and P. Kim, Convergence on error correction methods for solving initial value problems, J. Comput. Appl. Math., 236(17)(2012), 4448-4461. https://doi.org/10.1016/j.cam.2012.04.015
  17. P. Kim, E. Lee and S. D. Kim, Simple ECEM algorithms using function values only, Kyungpook Math. J., 53(2013), 573-591. https://doi.org/10.5666/KMJ.2013.53.4.573
  18. P. Kim and S. Bu, Error Control Strategy in Error Correction Methods, Kyungpook Math. J., 55(2015), 301-311. https://doi.org/10.5666/KMJ.2015.55.2.301
  19. G. Y. Kulikov and R. Weiner, Global error estimation and control in linearly-implicit parallel two-step peer W-methods, J. Comput. Appl. Math., 236(2011), 1226-1239. https://doi.org/10.1016/j.cam.2011.08.006
  20. L. F. Shampine, Error estimation and control for ODEs, J. Sci. Comput., 25(1)(2005), 3-16. https://doi.org/10.1007/s10915-004-4629-3
  21. L. F. Shampine, Vectorized solution of ODEs in MATLAB, Scalable Comput., Pract. Exp., 10(2010), 337-345.