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Stability properties of the Newmark, Houbolt and Wilson θ methods
Author(s) -
Gladwell Ian,
Thomas Ruth
Publication year - 1980
Publication title -
international journal for numerical and analytical methods in geomechanics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.419
H-Index - 91
eISSN - 1096-9853
pISSN - 0363-9061
DOI - 10.1002/nag.1610040205
Subject(s) - newmark beta method , stability (learning theory) , ordinary differential equation , scalar (mathematics) , mathematics , transformation (genetics) , differential equation , mathematical analysis , computer science , structural engineering , engineering , finite element method , geometry , biochemistry , chemistry , machine learning , gene
This paper analysis the stability of several methods for obtaining numerical solutions of second‐order ordinary differential equations. The methods are popular in structural and geotechnical engineering applications and are direct, that is they do not require the transformation of the second‐order equation into a first‐order system. They include Newmark's method in both implicit and explicit forms, Wilson's θ‐method, Houbolt's method and some variants on this latter method. We shall examine the stability of the methods when applied to the second‐order scalar test equation\documentclass{article}\pagestyle{empty}\begin{document}$$ \ddot x + 2a\dot x + (a^2 + c^2)x = 0 $$\end{document} where a and c are real .