NEW CONSTRUCTION OF HIGHER-ORDER LOCAL CONTINUOUS PLATFORMS FOR ERROR CORRECTION METHODS

Sunyoung Bu

doi:10.11948/2016033

2016 Volume 6 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2016 > 6(2): 443-462

Next Article Previous Article

Sunyoung Bu. NEW CONSTRUCTION OF HIGHER-ORDER LOCAL CONTINUOUS PLATFORMS FOR ERROR CORRECTION METHODS[J]. Journal of Applied Analysis & Computation, 2016, 6(2): 443-462. doi: 10.11948/2016033

Citation:

Sunyoung Bu. NEW CONSTRUCTION OF HIGHER-ORDER LOCAL CONTINUOUS PLATFORMS FOR ERROR CORRECTION METHODS[J]. Journal of Applied Analysis & Computation, 2016, 6(2): 443-462. doi: 10.11948/2016033

NEW CONSTRUCTION OF HIGHER-ORDER LOCAL CONTINUOUS PLATFORMS FOR ERROR CORRECTION METHODS

Sunyoung Bu

Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea

Fund Project:

Abstract

Abstract

Error correction method (ECM)[6,7] which has been recently developed, is based on the construction of a local approximation to the solution on each time step, and has the excellent convergence order O(h^2p+2), provided the local approximation has a local residual error O(h^p). In this paper, we construct a higher-order continuous local platform to develop higher-order semi-explicit one-step ECM for solving initial value time dependent differential equations. It is shown that special choices of parameters for the local platform can lead to the improvement of the well-known explicit fourth and fifth order Runge-Kutta methods. Numerical experiments demonstrate the theoretical results.
- Error correction method /
- local platform /
- stiff initial value problem /
- Runge Kutta methods
MSC: 34A45;65L04;65L20;65L70

FullText(HTML)

References

[1]	https://www.dm.uniba.it/testset/testsetivpsolvers. Google Scholar
[2]	R.R. Ahmad, N. Yaacob and A.H. Mohd Murid, Explicit methods in solving stiff ordinary differential equations, Int. J. Comput. Math., 81(2004), 1407-1415. Google Scholar
[3]	J. Áand lverez, J. Rojo, An improved class of generalized Runge-Kutta methods for stiff problems, Part I:The scalar case, Appl. Math. Comput., 130(2002), 537-560. Google Scholar
[4]	C. W. Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, 1971. Google Scholar
[5]	E. Hairer and G. Wanner, Solving ordinary differential equations, Ⅱ Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics, Springer, 1996. Google Scholar
[6]	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. Google Scholar
[7]	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(2012)(17), 4448-4461. Google Scholar
[8]	S.D. Kim and P. Kim, Exponentially fitted Error Correction Methods for solving Initial Value Problems, Kyungpook Math. J., 52(2012), 167-177. Google Scholar
[9]	P. Kim, S.D. Kim and E. Lee, Simple ECEM algorithms using function values only, Kyungpook Math. J., 53(2013)(4), 573-591. Google Scholar
[10]	W. Liniger and R.A. Willoughby, Efficient integration methods for stiff systems of ordinary differential equations, SIAM J. Numer. Anal., 7(1970), 47-65. Google Scholar
[11]	H. Ramos and J. Vigo-Aguiar, A fourth-order Runge-Kutta method based on BDF-type Chebyshev approximations, J. Comp. Appl. Numer., 204(2007), 124-136. Google Scholar
[12]	L.F. Shampine, Vectorized solution of ODEs in MATLAB, Scalable Comput.:Pract. Experience, 10(2010), 337-345. Google Scholar