On some one-step implicit methods as dynamical systems

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...

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Bibliographic Details
Published inJournal of numerical analysis and approximation theory Vol. 32; no. 2
Main Author Călin-Ioan Gheorghiu
Format Journal Article
LanguageEnglish
Published Publishing House of the Romanian Academy 01.08.2003
Subjects
continuous dynamical systems
convergence
Newton method
one-step implicit methods
shadowing
stability
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Summary: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.
ISSN:2457-6794
2501-059X
DOI:10.33993/jnaat322-745