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Extracting macroscopic dynamics: model problems and algorithms

Givon, Dror and Kupferman, Raz and Stuart, Andrew (2004) Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity, 17 (6). R55-R127. ISSN 0951-7715. doi:10.1088/0951-7715/17/6/R01.

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In many applications, the primary objective of numerical simulation of time-evolving systems is the prediction of coarse-grained, or macroscopic, quantities. The purpose of this review is twofold: first, to describe a number of simple model systems where the coarse-grained or macroscopic behaviour of a system can be explicitly determined from the full, or microscopic, description; and second, to overview some of the emerging algorithmic approaches that have been introduced to extract effective, lower-dimensional, macroscopic dynamics. The model problems we describe may be either stochastic or deterministic in both their microscopic and macroscopic behaviour, leading to four possibilities in the transition from microscopic to macroscopic descriptions. Model problems are given which illustrate all four situations, and mathematical tools for their study are introduced. These model problems are useful in the evaluation of algorithms. We use specific instances of the model problems to illustrate these algorithms. As the subject of algorithm development and analysis is, in many cases, in its infancy, the primary purpose here is to attempt to unify some of the emerging ideas so that individuals new to the field have a structured access to the literature. Furthermore, by discussing the algorithms in the context of the model problems, a platform for understanding existing algorithms and developing new ones is built.

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Additional Information:© 2004 IOP Publishing Ltd and London Mathematical Society. Received 5 June 2003; In final form 7 April 2004; Published 20 August 2004. This paper is based on the 2002 Ron Diperna Memorial Lecture, given by AMS at the Mathematics Department, University of California, Berkeley, 7 February 2002. The authors are grateful to Xinyu He for helping with some preliminary numerical calculations and to Zvi Artstein, Alexandre Chorin, Robert Krasny, Christian Lubich, Weinan E, Greg Pavliotis, Sebastian Reich, Christof Schütte, Paul Tupper and Petter Wiberg for helpful comments on a preliminary draft. The authors are particularly grateful to Eric Vanden Eijnden who made a number of significant suggestions about how to structure the paper. DG and RK are supported in part by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities, and by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the US Department of Energy under Contract DE-AC03-76-SF00098. AMS is supported by the EPSRC (UK).
Funding AgencyGrant Number
Israel Science FoundationUNSPECIFIED
Department of Energy (DOE)DE-AC03-76-SF00098
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
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Other Numbering System NameOther Numbering System ID
Andrew StuartJ59
Issue or Number:6
Classification Code:PACS numbers: 05.45.−a, 05.10.−a
Record Number:CaltechAUTHORS:20170609-153344149
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Official Citation:Dror Givon et al 2004 Nonlinearity 17 R55
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78071
Deposited By: Tony Diaz
Deposited On:09 Jun 2017 23:27
Last Modified:15 Nov 2021 17:36

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