On some one-step implicit methods as dynamical systems


  • Călin-Ioan Gheorghiu Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy, Romania


one-step implicit methods, Newton method, continuous dynamical systems, convergence, stability, shadowing
Abstract views: 211


The one-step implicit methods, the backward Euler being the most known, require the solution of a nonlinear equation at each step.To avoid this, these methods can be approximated by making use of a one step of a Newton method. Thus the methods are transformed into some explicit ones. We will obtain these transformed methods, find conditions under which they generate continuous dynamical systems and show their order of convergence. Some results on the stability of these explicit schemes, as well as on the shadowing phenomenon are also carried out. Concluding remarks and some open problems end the paper.


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Arveson, W., A Short Course in Spectral Theory, Springer, 2002.

Conway, J. B., A Course in Functional Analysis, Springer, 1985.

van Dorsselaer, J. L. M., Kraaijvanger, J. F. B. M. and Spijker, M. N., Linear stability analysis in numerical solutions of initial value problems, Acta Numerica, pp. 199-237, 1993, https://doi.org/10.1017/S0962492900002361.

Embree, M. and Trefethen, L. N., Pseudospectra Gateway, http://www.comlab.ox.ac.uk/projects/pseudospectra2000.

Gheorghiu, C. I., On the measure of non-normality of matrices, General Mathematics, ULB Sibiu, 2003.

Henrici, P., Discrete Variable Methods in Ordinary Differential Equations, John Wiley & Sons, 1962.

Kantorovich, L. V. and Akilov, G. P., Functional Analysis, Scientific Publishing House, Bucharest, Romania (translation from Russian).

Richtmayer, R. D. and Morton, K. W., Difference Methods for Initial-Value Problems, John Wiley, New York, 1967.

Stuart, A. M. and Humphries, A. R., Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996.

Süli, E. and Mayers, D., Numerical Computation I and II, Oxford University Computing Laboratory, Oxford, 1995.

Trefethen, L. N., Computation of Psedudospectra, Acta Numerica, pp. 247-295, 1999, https://doi.org/10.1017/S0962492900002932.

Trefethen, L. N. and Bau III, D., Numerical Linear Algebra, SIAM, Philadelphia, 1997.




How to Cite

Gheorghiu, C.-I. (2003). On some one-step implicit methods as dynamical systems. Rev. Anal. Numér. Théor. Approx., 32(2), 171–175. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/2003-vol32-no2-art5