DMCA
Deciding the nature of the coarse equation through microscopic simulations: the Baby-Bathwater scheme
| Venue: | SIAM MMS |
| Citations: | 23 - 10 self |
BibTeX
@ARTICLE{Li_decidingthe,
author = {Ju Li and Panayotis G. Kevrekidis and C. William Gear},
title = {Deciding the nature of the coarse equation through microscopic simulations: the Baby-Bathwater scheme},
journal = {SIAM MMS},
year = {},
pages = {556--574}
}
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Abstract
Abstract. 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.
Keyphrases
coarse equation microscopic simulation baby-bathwater scheme microscopic simulator periodic boundary condition sample variance explicit formula closed form coarse dynamic random initial condition certain statistic effective coupling macrosocopic behavior similar question spatial derivative certain decision plausibility test expected macroscopic behavior certain coarse conservation law recent development boundary condition certain conservation law multiscale computation initialized burst unavailable coarse equation equation-free approach important part simple scheme good solution practice coarse hamiltonian chosen family
