A numerical method for the solution of an autonomous initial value problem

Abstract

Using a known interpolation formula we introduce a class of numerical methods for approximating the solutions of scalar initial value problems for first order differential equations, which can be identified as explicit Runge-Kutta methods. We determine bounds for the local truncation error and we also compare the convergence order and the stability region with those for explicit Runge-Kutta methods, which have convergence order equal with number of stages (i.e. with 2, 3 and 4 stages). The convergence order is only two, but our methods have a larger absolute stability region than the above mentioned methods. In the last section a numerical example is provided, and the obtained numerical approximation is compared with the corresponding exact solution.

Authors

Flavius Patrulescu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Keywords

initial value problem; stability region; convergence order; local truncation error

Cite this paper as

F. Pătrulescu, A numerical method for the solution of an autonomous initial value problem, Carpathian J. Math. vol. 28 (2012), pp. 289-296

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About this paper

Journal

Carpathian Journal of Mathematics

Publisher Name

North University of Baia Mare, Department of Mathematics and Computer Science, Baia Mare

Print ISSN

1584-2851

Online ISSN

1843-4401

MR

3027258

ZBL

1289.65153

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References

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References

[1] Crouzeix, A.L. Mignot, Analyse numerique des equations differentielles, Masson, Paris (1989).
[2] D. Lambert, Numerical Methods for Ordinary Differential Systems. The Initial Value Problem, John Wiley & Sons, Chichester (1990).
[3] Pavaloiu, On an approximation formula, Rev. Anal. Numer. Theor. Approx.,  26, no. 1-2 (1997), 179-183.
[4] Ralston, Runge-Kutta Methods with Minimum Error Bounds, Math. Comp.,16, no. 80 (1962), 431-437.
[5] F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1964).

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