dorsal/arxiv
View SchemaDeciding the Nature of the "Coarse Equation" through Microscopic Simulations: the Baby-Bathwater Scheme
| Authors | Ju Li, Panayotis G. Kevrekidis, C. William Gear, Ioannis G. Kevrekidis |
|---|---|
| Categories | |
| ArXiv ID | physics/0212034 |
| URL | https://arxiv.org/abs/physics/0212034 |
Abstract
Recent developments in multiscale computation allow the solution of ``coarse equations'' for the expected macroscopic behavior of microscopically/stochastically evolving particle distributions 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 embodied in this approach requires certain decisions about the nature of the unavailable ``coarse equation''. Such decisions include (a) the determination of the highest spatial derivative active in the equation, (b) whether the coarse 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 nature of 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. Microscopic 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 argue that such schemes constitute an important part of the equation-free approach to multiscale computation.
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"abstract": "Recent developments in multiscale computation allow the solution of ``coarse\nequations\u0027\u0027 for the expected macroscopic behavior of\nmicroscopically/stochastically evolving particle distributions without ever\nobtaining these coarse equations in closed form. The closure is obtained ``on\ndemand\u0027\u0027 through appropriately initialized bursts of microscopic simulation.\nThe effective coupling of microscopic simulators with macrosocopic behavior\nembodied in this approach requires certain decisions about the nature of the\nunavailable ``coarse equation\u0027\u0027. Such decisions include (a) the determination\nof the highest spatial derivative active in the equation, (b) whether the\ncoarse equation satisfies certain conservation laws, and (c) whether the coarse\ndynamics is Hamiltonian. These decisions affect the number and type of boundary\nconditions as well as the nature of the algorithms employed in good solution\npractice. In the absence of an explicit formula for the temporal derivative, we\npropose, implement and validate a simple scheme for deciding these and other\nsimilar questions about the coarse equation using only the microscopic\nsimulator. Microscopic simulations under periodic boundary conditions are\ncarried out for appropriately chosen families of random initial conditions;\nevaluating the sample variance of certain statistics over the simulation\nensemble allows us to infer the highest order of spatial derivatives active in\nthe coarse equation. In the same spirit we show how to determine whether a\ncertain coarse conservation law exists or not, and we discuss plausibility\ntests for the existence of a coarse Hamiltonian or integrability. We argue that\nsuch schemes constitute an important part of the equation-free approach to\nmultiscale computation.",
"arxiv_id": "physics/0212034",
"authors": [
"Ju Li",
"Panayotis G. Kevrekidis",
"C. William Gear",
"Ioannis G. Kevrekidis"
],
"categories": [
"physics.comp-ph"
],
"title": "Deciding the Nature of the \"Coarse Equation\" through Microscopic Simulations: the Baby-Bathwater Scheme",
"url": "https://arxiv.org/abs/physics/0212034"
},
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