TY - JOUR

T1 - Deciding the nature of the coarse equation through microscopic simulations

T2 - The baby-bathwater scheme

AU - Li, Ju

AU - Kevrekidis, Panayotis G.

AU - Gear, C. William

AU - Kevrekidis, Ioannis G.

N1 - Funding Information:
This work was partially supported by Honda R and D Co., Ltd. and the OSU Transportation Research Endowment Program (JL), the National Science Foundation (IGK, PGK), AFOSR (CWG, IGK) and the Clay Institute (PGK).
Funding Information:
∗Received by the editors December 7, 2002; accepted for publication (in revised form) May 13, 2003; published electronically July 17, 2003. This work was partially supported by Honda R&D Co., Ltd. and the OSU Transportation Research Endowment Program (JL), the National Science Foundation (IGK, PGK), AFOSR (CWG, IGK) and the Clay Institute (PGK). http://www.siam.org/journals/mms/1-3/41916.html †Department of Materials Science and Engineering, Ohio State University, Columbus, OH 43210 (lij@mse.eng.ohio-state.edu). ‡Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515 (kevrekidis@math.umass.edu). §NEC Research Institute, 4 Independence Way, Princeton, NJ 08540 (cwg@nec-labs.com). ¶Department of Chemical Engineering and PACM, Princeton University, Princeton, NJ 08544 (yannis@princeton.edu).
Publisher Copyright:
©2003 Society for Industrial and Applied Mathematics.

PY - 2003

Y1 - 2003

N2 - Recent developments in multiscale computation allow the solution of coarse equations for the expected macroscopic behavior of microscopically evolving particles without ever obtaining these coarse equations in closed form. The closure is obtained on demand through appropriately initialized bursts of microscopic simulation. The effective coupling of microscopic simulators with macrosocopic behavior requires certain decisions about the nature of the unavailable coarse equation. Such decisions include (a) the highest spatial derivative active in the coarse equation, (b) whether the equation satisfies certain conservation laws, and (c) whether the coarse dynamics is Hamiltonian. These decisions affect the number and type of boundary conditions as well as the algorithms employed in good solution practice. In the absence of an explicit formula for the temporal derivative, we propose, implement, and validate a simple scheme for deciding these and other similar questions about the coarse equation using only the microscopic simulator. Simulations under periodic boundary conditions are carried out for appropriately chosen families of random initial conditions; evaluating the sample variance of certain statistics over the simulation ensemble allows us to infer the highest order of spatial derivatives active in the coarse equation. In the same spirit we show how to determine whether a certain coarse conservation law exists or not, and we discuss plausibility tests for the existence of a coarse Hamiltonian or integrability. We believe that such schemes constitute an important part of the equation-free approach to multiscale computation.

AB - Recent developments in multiscale computation allow the solution of coarse equations for the expected macroscopic behavior of microscopically evolving particles without ever obtaining these coarse equations in closed form. The closure is obtained on demand through appropriately initialized bursts of microscopic simulation. The effective coupling of microscopic simulators with macrosocopic behavior requires certain decisions about the nature of the unavailable coarse equation. Such decisions include (a) the highest spatial derivative active in the coarse equation, (b) whether the equation satisfies certain conservation laws, and (c) whether the coarse dynamics is Hamiltonian. These decisions affect the number and type of boundary conditions as well as the algorithms employed in good solution practice. In the absence of an explicit formula for the temporal derivative, we propose, implement, and validate a simple scheme for deciding these and other similar questions about the coarse equation using only the microscopic simulator. Simulations under periodic boundary conditions are carried out for appropriately chosen families of random initial conditions; evaluating the sample variance of certain statistics over the simulation ensemble allows us to infer the highest order of spatial derivatives active in the coarse equation. In the same spirit we show how to determine whether a certain coarse conservation law exists or not, and we discuss plausibility tests for the existence of a coarse Hamiltonian or integrability. We believe that such schemes constitute an important part of the equation-free approach to multiscale computation.

KW - Coarse dynamics

KW - Equation-free

KW - Identification

KW - Microscopic simulator

KW - Multiscale computation

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U2 - 10.1137/S1540345902419161

DO - 10.1137/S1540345902419161

M3 - Article

AN - SCOPUS:85021663679

SN - 1540-3459

VL - 1

SP - 391

EP - 407

JO - Multiscale Modeling and Simulation

JF - Multiscale Modeling and Simulation

IS - 3

ER -