[344b] - 'Coarse' Stability/Bifurcation Analysis of Monte-Carlo
Reaction Simulations
Author
Information:
- Dimitrios
Maroudas
- University of California-Santa
Barbara
- Department of Chemical
Engineering
- Santa Barbara, CA
93106-5080
- Phone:
805-893-7346
- Fax: 805-893-4731
- Email: maroudas@engineering.ucsb.edu
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- Yannis
Kevrekidis (speaker)
- Princeton
University
- Room A-207 Engineering
Quadrangle
- Princeton, NJ
08544
- Phone:
609-258-2818
- Fax: 609-258-0211
- Email: yannis@Princeton.edu
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- Saravanapriyan
Sriraman
- University of
California
- Department of Chemical
Engineering
- Santa Barbara, CA
93106
- Phone:
805-893-8151
- Fax: 805-893-473`
- Email: rudram@engineering.ucsb.edu
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- Alexei Makeev
- Moscow State
University
- Vorobjovy Gory
V-234
- Moscow, 119899
- Russia
- Phone: +7(095)
939-4079
- Fax:
- Email: amak@redsun.cs.msu.edu
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Abstract:
Stochastic microscopic simulations, such
as those employing the Monte Carlo method, are often the tool of
choice in simulating the behavior of chemically reacting systems,
and, in particular, of surface catalytic reactions. While such codes
have been successful in analyzing and understanding the nonlinear
behavior of mesoscopic and macroscopic quantities (local adsorbate
coverages, overall reaction rates), they are not in principle
capable of systematic stability/bifurcation analysis at a "coarse"
engineering system level.
In this contribution we employ
methods motivated by continuum numerical bifurcation theory, and, in
particular, time-stepper based bifurcation algorithms, to perform
what we will term "coarse", "system-level" analysis of Monte Carlo
simulations of reacting systems. The word "coarse" is used since we
will study quantities such as average concentrations (moments of
distributions evolving through the MC simulation). The expression
"system-level" comes from the nature of the tasks we will perform:
parametric continuation of stationary states, stability and
bifurcation analysis of the expected values of these moments with
respect to both molecular and system-level parameters. For a certain
class of problems, for which mesoscopic, closed evolution equations
for the expected values conceptually exist but are not available in
closed form, this approach may offer significant alternatives to
direct MC simulations.
The basis of the approach consists of
using a time-stepper for the expected values that is constructed not
from an approximate mesoscopic equation, which are assumed
unavailable in closed form, but through several microscopic
computational realizations of a finite time trajectory (that is
where the expression "time-stepper" comes from). We use this time
stepper to perform two types of tasks:
(a) coarse stability
and bifurcation analysis of surface catalytic reaction models based
on microscopic MC simulations - these include models of CO oxidation
and the NO+H2 reaction - exhibiting coarse steady-state multiplicity
and oscillations, that is, turning points and Hopf
bifurcations
(b) coarse integration, that is, evolution of
the expected values on mesoscopic time scales using recently
constructed projective integrators, as well as making use of many,
short, parallel, appropriately initialized microscopic
integrations.
Finally, we discuss the extension - to the
analysis of the MC simulations - of algorithms based on the
augmented continuum systems used for standard bifurcation analysis.
Both continuum-based and direct microscopic implementations of such
algorithms will be discussed. One of the items we will discuss is
the possible detection (by the algorithm) of instances in which one
level of mesoscopic closure becomes inaccurate, and another level of
mesoscopic closure (e.g., one involving two-point correlation
functions) may become necessary.
Overall, we believe that
this class of algorithms, targeting unavailable in closed form
mesoscopic descriptions, may prove a competitive computational
alternative to direct simulation in the study of MC - and other,
microscopic, such as MD and LB, or hybrid, like DSMC - simulations
of reaction/transport processes.
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Contact Information: