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Hierarchical higher‐order dissipative methods for transient analysis
Author(s) -
Govoni Laura,
Mancuso Massimo,
Ubertini Francesco
Publication year - 2006
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.1682
Subject(s) - spurious relationship , discontinuity (linguistics) , transient (computer programming) , estimator , dissipative system , computer science , mathematics , collocation (remote sensing) , mathematical optimization , hierarchy , algorithm , work (physics) , mathematical analysis , engineering , statistics , physics , quantum mechanics , machine learning , economics , market economy , operating system , mechanical engineering
This work focuses on devising an efficient hierarchy of higher‐order methods for linear transient analysis, equipped with an effective dissipative action on the spurious high modes of the response. The proposed strategy stems from the Nørsett idea and is based on a multi‐stage algorithm, designed to hierarchically improve accuracy while retaining the desired dissipative behaviour. Computational efficiency is pursued by requiring that each stage should involve just one set of implicit equations of the size of the problem to be solved (as standard time integration methods) and, in addition, all the stages should share the same coefficient matrix. This target is achieved by rationally formulating the methods based on the discontinuous collocation approach. The resultant procedure is shown to be well suited for adaptive solution strategies. In particular, it embeds two natural tools to effectively control the error propagation. One estimates the local error through the next‐stage solution, which is one‐order more accurate, the other through the solution discontinuity at the beginning of the current time step, which is permitted by the present formulation. The performance of the procedure and the quality of the two error estimators are experimentally verified on different classes of problems. Some typical numerical tests in transient heat conduction and elasto‐dynamics are presented. Copyright © 2006 John Wiley & Sons, Ltd.