Time-averaged coarse variables for multi-scale dynamics
Authors:
Marshall Slemrod and Amit Acharya
Journal:
Quart. Appl. Math. 70 (2012), 793-803
MSC (2010):
Primary 34E13; Secondary 34E15, 34C29, 35A35
DOI:
https://doi.org/10.1090/S0033-569X-2012-01291-5
Published electronically:
July 18, 2012
MathSciNet review:
3052092
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Additional Information
Abstract: Given an autonomous system of Ordinary Differential Equations without an a priori split into slow and fast components, we define a strategy for producing a large class of ‘slow’ variables (constants of fast motion) in a precise sense. The equation of evolution of any such slow variable is deduced. The strategy is to rewrite our system on an infinite-dimensional “history” Hilbert space $X$ and define our coarse observation as a functional on $X$.
References
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- Amit Acharya, Coarse-graining autonomous ODE systems by inducing a separation of scales: practical strategies and mathematical questions, Math. Mech. Solids 15 (2010), no. 3, 342–352. MR 2667382, DOI https://doi.org/10.1177/1081286508100972
- Amit Acharya and Aarti Sawant, On a computational approach for the approximate dynamics of averaged variables in nonlinear ODE systems: toward the derivation of constitutive laws of the rate type, J. Mech. Phys. Solids 54 (2006), no. 10, 2183–2213. MR 2254208, DOI https://doi.org/10.1016/j.jmps.2006.03.007
- Zvi Artstein, On singularly perturbed ordinary differential equations with measure-valued limits, Proceedings of EQUADIFF, 10 (Prague, 2001), 2002, pp. 139–152. MR 1981520
- Z. Artstein, C.W. Gear, I.G. Kevrekidis, M. Slemrod, and E.S. Titi, Analysis and computation of a discrete KdV-Burgers type equation with fast dispersion and slow diffusion, Arxiv preprint arXiv:0908.2752; to appear in SIAM Journal of Numerical Analysis (2009).
- Zvi Artstein, Ioannis G. Kevrekidis, Marshall Slemrod, and Edriss S. Titi, Slow observables of singularly perturbed differential equations, Nonlinearity 20 (2007), no. 11, 2463–2481. MR 2361254, DOI https://doi.org/10.1088/0951-7715/20/11/001
- Zvi Artstein and Alexander Vigodner, Singularly perturbed ordinary differential equations with dynamic limits, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), no. 3, 541–569. MR 1396278, DOI https://doi.org/10.1017/S0308210500022903
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
- Weinan E, HMM and the “Equation-free” approach to multiscale modeling, http://www.math.princeton.edu/~weinan/hmm.html.
- Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913
References
- A. Acharya, On the choice of coarse variables for dynamics, International Journal for Multiscale Computational Engineering 5 (2007), no. 6, 483–489.
- ---, Coarse-graining autonomous ODE systems by inducing a separation of scales: practical strategies and mathematical questions, Mathematics and Mechanics of Solids 15 (2010), no. 3, 342–352. MR 2667382 (2011d:34109)
- A. Acharya and A. Sawant, On a computational approach for the approximate dynamics of averaged variables in nonlinear ODE systems: Toward the derivation of constitutive laws of the rate type, Journal of the Mechanics and Physics of Solids 54 (2006), no. 10, 2183–2213. MR 2254208 (2007f:74044)
- Z. Artstein, On singularly perturbed ordinary differential equations with measure-valued limits, Mathematica Bohemica 127 (2002), no. 2, 139–152. MR 1981520 (2004d:34119)
- Z. Artstein, C.W. Gear, I.G. Kevrekidis, M. Slemrod, and E.S. Titi, Analysis and computation of a discrete KdV-Burgers type equation with fast dispersion and slow diffusion, Arxiv preprint arXiv:0908.2752; to appear in SIAM Journal of Numerical Analysis (2009).
- Z. Artstein, I.G. Kevrekidis, M. Slemrod, and E.S. Titi, Slow observables of singularly perturbed differential equations, Nonlinearity 20 (2007), 2463–2481. MR 2361254 (2008j:34087)
- Z. Artstein and A. Vigodner, Singularly perturbed ordinary differential equations with dynamic limits, Royal Society (Edinburgh), Proceedings, Section A 126 (1996), 541–569. MR 1396278 (97g:34073)
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, vol. 44, Springer, 1983. MR 710486 (85g:47061)
- Weinan E, HMM and the “Equation-free” approach to multiscale modeling, http://www.math.princeton.edu/~weinan/hmm.html.
- K. Yosida, Functional analysis, $6^{th}$ ed., Springer, 1980. MR 617913 (82i:46002)
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Additional Information
Marshall Slemrod
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706
MR Author ID:
163635
Email:
slemrod@math.wisc.edu
Amit Acharya
Affiliation:
Civil & Environmental Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvannia 15213
Email:
acharyaamit@cmu.edu
Received by editor(s):
August 29, 2011
Published electronically:
July 18, 2012
Article copyright:
© Copyright 2012
Brown University
The copyright for this article reverts to public domain 28 years after publication.
