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Multiscale MethodsBridging the Scales in Science and Engineering$
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Jacob Fish

Print publication date: 2009

Print ISBN-13: 9780199233854

Published to Oxford Scholarship Online: February 2010

DOI: 10.1093/acprof:oso/9780199233854.001.0001

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Principles of systematic upscaling

Principles of systematic upscaling

(p.193) 7 Principles of systematic upscaling
Multiscale Methods

Achi Brandt

Oxford University Press

Building on the complementary advantages of Renormalization Group (RG) and multigrid (MG) methods, Systematic Upscaling (SU) comprises rigorous procedures for deriving suitable variables and corresponding numerical equations (or statistical relations) that describe a given physical system at progressively larger scales, starting at some fine scale where the physical laws are known (in the form of a partial differential system, or a statistical-mechanics Hamiltonian, or Newton laws for moving particles, etc.). Unlike RG, the SU algorithms include repeated coarse-to-fine transitions, which are essential for (1) testing the adequacy of the set of coarse-level variables (thus providing a general tool for constructing that set); (2) accelerating the finer-level simulations; and, most importantly (3) confining those simulations to small representative subdomains. No substantial scale separation is assumed; as in MG, small scale ratio between successive levels is in fact important to ensure slowdown-free simulations at all scales. Detailed examples are given in terms of local-interaction systems at equilibrium, and extensions are briefly discussed to long-range interactions, dynamic systems, low temperatures, and more.

Keywords:   upscaling, renormalization, multigrid, multiscale, coarsening, local refinement, relaxation, Monte Carlo, molecular dynamics, polymer

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