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High‐order integration of smooth dynamical systems: Theory and numerical experiments
Author(s) -
Austin Mark
Publication year - 1993
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620361210
Subject(s) - extrapolation , newmark beta method , mathematics , numerical integration , dynamical systems theory , order (exchange) , asymptotic expansion , numerical analysis , linear system , work (physics) , class (philosophy) , richardson extrapolation , algorithm , mathematical analysis , mathematical optimization , nonlinear system , computer science , engineering , physics , mechanical engineering , finance , quantum mechanics , artificial intelligence , economics
This paper describes a new class of algorithms for integrating linear second‐order equations and those containing smooth non‐linearities. The algorithms are based on a combination of ideas from standard Newmark integration methods and extrapolation techniques. For the algorithm to work, the underlying Newmark method must be stable, second‐order accurate, and produce asymptotic error expansions for response quantities containing only even‐ordered terms. It is proved that setting the Newmark parameter γ equal to 1/2 gives a desirable asymptotic expansion, irrespective of the setting for β. Numerical experiments are conducted for one linear and two non‐linear applications.