# On some one-step implicit methods as dynamical systems

## Abstract

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.

## Authors

Călin-Ioan Gheorghiu
Tiberiu Popoviciu Institute of Numerical Analysis

## Keywords

one-step implicit methods; Newton method; continuous dynamical systems; convergence; stability; shadowing.

## Paper coordinates

C.I. Gheorghiu, On some one-step implicit methods as dynamical systems, Rev. Anal. Numér. Théor. Approx., 32 (2003) 171-175.

## PDF

##### Journal

Rev. Anal. Numér. Théor. Approx

1222-9024

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