Kyungpook Mathematical Journal

  • 제55권2호
  • /
  • Pages.301-311
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  • 2015
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  • 1225-6951(pISSN)
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  • 0454-8124(eISSN)
경북대학교 자연과학대학 수학과 (Department of Mathematics, Kyungpook National University)
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Error Control Strategy in Error Correction Methods

  • KIM, PHILSU (Department of Mathematics, Kyungpook National University) ;
  • BU, SUNYOUNG (Institute for Mathematical Convergence, Kyungpook National University)
  • 투고 : 2015.01.27
  • 심사 : 2015.04.09
  • 발행 : 2015.06.23
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초록

In this paper, we present the error control techniques for the error correction methods (ECM) which is recently developed by P. Kim et al. [8, 9]. We formulate the local truncation error at each time and calculate the approximated solution using the solution and the formulated truncation error at previous time for achieving uniform error bound which enables a long time simulation. Numerical results show that the error controlled ECM provides a clue to have uniform error bound for well conditioned problems [1].

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참고문헌

  1. K. E. Atkinson, An introduction to numerical analysis, John Wiley & Sons, Inc. (1989).
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  3. S. Bu, J. Huang, and M. M. Minion, Semi-implicit Krylov Deferred Correction Methods for Differential Algebraic Equations, Mathematics of Computation, 81(2012), 2127-2157. https://doi.org/10.1090/S0025-5718-2012-02564-6
  4. S. Bu, J. Lee, An enhanced parareal algorithm based on the deferred correction methods for a stiff system, Journal of Computational and Applied Mathematics, 255(1)(2014), 297-305. https://doi.org/10.1016/j.cam.2013.05.001
  5. E. Hairer, G. Wanner, Solving ordinary differential equations. II Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics, Springer (1996).
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  7. 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
  8. 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
  9. S. D. Kim, X. Piao, D. H. Kim and P. Kim, Convergence on error correction methods for solving initial value problems, J. Comp. Appl. Math., 236(17)(2012), 4448-4461. https://doi.org/10.1016/j.cam.2012.04.015
  10. P. Kim, S. D. Kim and E. Lee, Simple ECEM algorithms using function values only, Kyungpook Mathematical Journal, 53(4)(2013), 573-591. https://doi.org/10.5666/KMJ.2013.53.4.573
  11. G. Y. Kulikov, Global Error Control in Adaptive Nordsieck Methods, SIAM J. Sci. Comput., 34(2)(2012), A839-A860. https://doi.org/10.1137/100791932
  12. H. Ramos, A non-standard explicit integration scheme for initial-value problems, Appl. Math. Comp., 189(1)(2007), 710-718. https://doi.org/10.1016/j.amc.2006.11.134

피인용 문헌

  1. An Error Embedded Runge-Kutta Method for Initial Value Problems vol.56, pp.2, 2016, https://doi.org/10.5666/KMJ.2016.56.2.311