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Interval schemes for singularly perturbed initial value problems. (English) Zbl 1073.65056

Summary: The modified exponential interval schemes are introduced for the solution of singularly perturbed initial value problems. We give the outline of constructing the schemes of \(k\)-th order, then we construct four schemes for \(k = 1\) and \(k = 2\). These schemes are uniformly convergent of second and third order accuracy. Also, we introduce the idea of optimal convergence. Numerical results and comparisons with other schemes are presented.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34E15 Singular perturbations for ordinary differential equations
65G40 General methods in interval analysis
65L20 Stability and convergence of numerical methods for ordinary differential equations
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[1] Alefeld, G. and Herzberger, J.: Introduction to Interval Computations, Academic Press, New York, 1983. · Zbl 0552.65041
[2] Carroll, J.: A Matricial Exponentially Fitted Scheme for the Numerical Solution of Stiff Initial-Value Problems, Computers Math. Applic. 26, 1993, pp. 57-64. · Zbl 0789.65055 · doi:10.1016/0898-1221(93)90034-S
[3] Carroll, J.: A Uniformly Convergent Exponentially Fitted DIRK Scheme, in: Miller, J. J. H. (ed.), Proc. BAIL III Conf. on Computational and Asymptotic Methods for Boundary and Initial Layers, Boole Press, Dublin, 1984. · Zbl 0672.65057
[4] Carroll, J.: On the Implementation of Exponentially Fitted One-Step Methods for the Numerical Integration of Stiff Linear Initial Value Problems, in: Miller, J. J. H. (ed.), Proc. BAIL II Conf. on Computational and Asymptotic Methods for Boundary and Initial Layers, Boole Press, Dublin, 1982. · Zbl 0511.65049
[5] Corliss, G. F.: Guaranteed Error Bounds for Ordinary Differential Equations, in: Ainsworth, M., Light, W. A., and Marletta, M. (eds): Theory and Numerics of Ordinary and Partial Differential Equations, Clarendon Press, Oxford, 1995, pp. 1-75, http://www.eng.mu.edu/corlissg/. · Zbl 0843.65060
[6] Corliss, G. F. and Rihm, R.: Validating an A Priori Enclosure Using High-Order Taylor Series, in: Alefeld, G., Frommer, A., and Lang, B. (eds), Scientific Computing, Computer Arithmetic, and Validated Numerics, Akademie Verlag, Berlin, 1996, pp. 228-238. · Zbl 0851.65054
[7] Doolan, E. P., Miller, J. J. H., and Schilders, W. H. A.: Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, 1980. · Zbl 0459.65058
[8] Farrell, P. A.: Uniform and Optimal Schemes for Stiff Initial-Value Problems, Computers Math. Applic. 13 (1987), pp. 925-936. · Zbl 0632.65084 · doi:10.1016/0898-1221(87)90065-4
[9] Keller, H. B.: Numerical Methods for Two-Point Boundary Value Problems, Blaisdell Publishing Company, 1968. · Zbl 0172.19503
[10] Lohner, R. J.: Einschlie?ung der L?sung gew?hnlicher Anfangs-und Randwertaufgaben und Anwendungen, Ph.D. thesis, Universit?t Karlsruhe, 1988. · Zbl 0663.65074
[11] Lohner, R. J.: Enclosures for the Solutions of Ordinary Initial and Boundary Value Problems, in: Cash, J. R. and Gladwell, I. (eds), Computational Ordinary Differential Equations, Clarendon Press, Oxford, 1992, pp. 425-435. · Zbl 0767.65069
[12] Lohner, R. J.: Step Size and Order Control in the Verified Solution of IVP with ODEs, in: SciCADE?95 International Conference on Scientific Computation and Differential Equations, Stanford, California, 1995.
[13] Miller, J. J. H.: Optimal Uniform Difference Schemes for Linear Initial-Value Problems, Computer Math. Applic. 12B (1986), pp. 1209-1215. · Zbl 0659.65072 · doi:10.1016/0898-1221(86)90245-2
[14] Moore, R. E.: Automatic Local Coordinate Transformations to Reduce the Growth of Error Bounds in Interval Computation of Solution of Ordinary Differential Equations, in: Rall, L. B. (ed.), Error in DigitalComputation II, John Wiley, New York, 1965, pp. 103-140. · Zbl 0202.45101
[15] Moore, R. E.: Interval Analysis, Prentice Hall, Englewood Cliffs, 1966. · Zbl 0176.13301
[16] Moore, R. E.: The Automatic Analysis and Control of Error in Digital Computation Based on the Use of Interval Numbers, in: Rall, L. B. (ed.), Error in Digital Computation I, John Wiley, NewYork, 1965, pp. 61-130. · Zbl 0202.45002
[17] Nedialkov, N. S.: Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation, Ph.D. thesis, Computer Science Dept., Toronto, 1999, http://www.cs.toronto.edu/NA/reports.html. · Zbl 0947.65081
[18] Neher, M.: An Enclosure Method for the Solution of Linear ODEs with Polynomial Coefficients, Numer. Funct. Anal. Optimiz. 20 (1999), pp. 779-803. · Zbl 0936.65084 · doi:10.1080/01630569908816923
[19] Neher, M.: Geometric Series Bounds for the Local Errors of Taylor Methods for Linear n-th Order ODEs, in: Alefeld, G., Rohn, J., Rump, S., and Yamamoto, T. (eds), Symbolic Algebraic Methods and Verification Methods, Springer, Wien, 2001, pp. 183-193. · Zbl 0987.65064
[20] Rihm, R.: Interval Methods for Initial Value Problems in ODEs, in: Herzberger, J. (ed.), Topics in Validated Computations, Elsevier, Amsterdam, 1994, pp. 173-208. · Zbl 0815.65095
[21] Rump, S. M.: INTerval LABoratory, in: Csendes, T. (ed.), Developments in Reliable Computing, Kluwer, Dordrecht, 1999, pp. 77-105, http://www.ti3.tu-harburg.de/\(\sim\)rump/intlab/. · Zbl 0949.65046
[22] Salama, A. A.: A Class of Exponential Methods for Stiff Initial-Value Problems, Intern. J. Computer Math. 62 (1996), pp. 183-198. · Zbl 0870.65056 · doi:10.1080/00207169608804536
[23] Salama, A. A.: Higher Order and Optimal Schemes for Stiff Initial-Value Problems, Math. Japonica 52 (2000), pp. 377-386. · Zbl 0974.65070
[24] Salama, A. A.: The Operator Compact Implicit Methods for the Numerical Treatment of Ordinary Differential and Integro-Differential Equation, Ph.D. thesis, Assiut University, Assiut, 1992.
[25] Stauning, O.: Automatic Validation of Numerical Solutions, Tech. report, IMM-PHD-1997-36, IMM, Lyngby, Denmark, 1997, http://www.imm.dtu.dk/fadbad.html/.
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