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Exponentially Fitted Error Correction Methods for Solving Initial Value Problems
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  • Journal title : Kyungpook mathematical journal
  • Volume 52, Issue 2,  2012, pp.167-177
  • Publisher : Department of Mathematics, Kyungpook National University
  • DOI : 10.5666/KMJ.2012.52.2.167
 Title & Authors
Exponentially Fitted Error Correction Methods for Solving Initial Value Problems
Kim, Sang-Dong; Kim, Phil-Su;
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In this article, we propose exponentially fitted error correction methods(EECM) which originate from the error correction methods recently developed by the authors (see [10, 11] for examples) for solving nonlinear stiff initial value problems. We reduce the computational cost of the error correction method by making a local approximation of exponential type. This exponential local approximation yields an EECM that is exponentially fitted, A-stable and L-stable, independent of the approximation scheme for the error correction. In particular, the classical explicit Runge-Kutta method for the error correction not only saves the computational cost that the error correction method requires but also gives the same convergence order as the error correction method does. Numerical evidence is provided to support the theoretical results.
Exponentially fitted;Error correction;Stiff initial-value problem;RK methods;
 Cited by
C. E. Abhulimen, G. E. Omeike, A sixth-order exponentially fitted shceme for the numerical solution of systems of ordinary differential equations, J. Appl. Math. & Bioinf., 1(2011), 175-186.

R. R. Ahmad, N. Yaacob, A. H. Mohd Murid, Explicit methods in solving stiff ordinary differential equations, Int. J. Comput. Maht., 81(2004), 1407-1415. crossref(new window)

J. Alverez, 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. crossref(new window)

L. Brugnano and C. Magherini, Blended implementation of block implicit methods for ODEs, Appl. Numer. Math., 42(2002), 29-45. crossref(new window)

J. R. Cash, On the exponential fitting of composite, multiderivative linear multistep methods, SIAM J. Numer. Anal., 18(1981), 808-821. crossref(new window)

M. V. Daele and G. V. Berghe, Extended one-step methods: an exponential fitting approach, Appl. Num. Anal. Comp. Math., 1(2004), 353-362. crossref(new window)

G. Dahlquist, A special stability problem for linear multistep methods, BIT, 3(1963), 27-43. crossref(new window)

L. W. Jackson and S. K. Kenue, A fourth order exponetially fitted method, SIAM J. Numer. Anal., 11(1974), 965-978. crossref(new window)

W. Liniger and R. A. Willoughby, Efficient integration methods for stiff systems of ordinary differential equations, SIAM J. Numer. Anal., 7(1970), 47-65. crossref(new window)

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. crossref(new window)

S. D. Kim, X. Piao and P. Kim, Convergence on Error correction methods for initial value problems, J. Comp. Appl. Math., to appear.

H. Ramos, A non-standard explicit integration scheme for initial-value problems, Appl. Math. Comp., 189(1)(2007), 710-718. crossref(new window)

D. Voss, A fifth-order exponentially fitted formula, SIAM J. Numer. Anal., 25(1988), 670-678. crossref(new window)

X. Y. Wu, A sixth-order A-stable explicit one-step method for stiff systems, Comput. Math. Appl., 35(1998), 59-64.