# Error Control Strategy in Error Correction Methods

• KIM, PHILSU (Department of Mathematics, Kyungpook National University) ;
• BU, SUNYOUNG (Institute for Mathematical Convergence, Kyungpook National University)
• Accepted : 2015.04.09
• Published : 2015.06.23

#### Abstract

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].

#### Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

#### References

1. K. E. Atkinson, An introduction to numerical analysis, John Wiley & Sons, Inc. (1989).
2. L. Brugnano and C. Magherini, Blended implementation of block implicit methods for ODEs, Appl. Numer. Math., 42(2002), 29-45. https://doi.org/10.1016/S0168-9274(01)00140-4
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).
6. K. Gustafsson, Control-theoretic techniques for stepsize selection in implicit Runge-Kutta methods, ACM Transactions on Mathematical Software, 20(4)(1994), 496-517. https://doi.org/10.1145/198429.198437
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.

#### Cited by

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