Noise propagation in hybrid models of nonlinear systems: The Ginzburg–Landau equation

Journal Article · Tue Apr 01 00:00:00 EDT 2014 · Journal of Computational Physics

DOI:https://doi.org/10.1016/J.JCP.2014.01.015· OSTI ID:22314856

Taverniers, Søren ^[1]; Alexander, Francis J. ^[2]; Tartakovsky, Daniel M. ^[1]

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1. Department of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA 92093 (United States)
2. Los Alamos National Laboratory, Los Alamos, NM 87545 (United States)

Every physical phenomenon can be described by multiple models with varying degrees of fidelity. The computational cost of higher fidelity models (e.g., molecular dynamics simulations) is invariably higher than that of their lower fidelity counterparts (e.g., a continuum model based on differential equations). While the former might not be suitable for large-scale simulations, the latter are not universally valid. Hybrid algorithms provide a compromise between the computational efficiency of a coarse-scale model and the representational accuracy of a fine-scale description. This is achieved by conducting a fine-scale computation in subdomains where it is absolutely required (e.g., due to a local breakdown of a continuum model) and coupling it with a coarse-scale computation in the rest of a computational domain. We analyze the effects of random fluctuations generated by the fine-scale component of a nonlinear hybrid on the hybrid's overall accuracy and stability. Two variants of the time-dependent Ginzburg–Landau equation (GLE) and their discrete representations provided by a nearest-neighbor Ising model serve as a computational testbed. Our analysis shows that coupling these descriptions in a one-dimensional simulation leads to erroneous results. Adding a random source term to the GLE provides accurate prediction of the mean behavior of the quantity of interest (magnetization). It also allows the two GLE variants to correctly capture the strength of the microscale fluctuations. Our work demonstrates the importance of fine-scale noise in hybrid simulations, and suggests the need for replacing an otherwise deterministic coarse-scale component of the hybrid with its stochastic counterpart.

• OSTI ID: 22314856
• Journal Information: Journal of Computational Physics, Vol. 262; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGORITHMS
COMPUTERIZED SIMULATION
COUPLING
FLUCTUATIONS
GINZBURG-LANDAU THEORY
ISING MODEL
MAGNETIZATION
MATHEMATICAL SOLUTIONS
MOLECULAR DYNAMICS METHOD
NONLINEAR PROBLEMS
ONE-DIMENSIONAL CALCULATIONS
PARTIAL DIFFERENTIAL EQUATIONS
RANDOMNESS
SCALE MODELS
STOCHASTIC PROCESSES
TIME DEPENDENCE