TY - JOUR

T1 - Coarse projective kMC integration

T2 - Forward/reverse initial and boundary value problems

AU - Rico-Martínez, R.

AU - Gear, C. W.

AU - Kevrekidis, Ioannis G.

N1 - Funding Information:
This work was partially supported by AFOSR (Dynamics and Control), an NSF-ITR Grant (I.G.K., C.W.G.) and Fulbright-Garcı́a Robles and CONACYT Fellowships (R.R.M.). Discussions with Profs. Li Ju, P.G. Kevrekidis, L. Petzold, and Dr. G. Hummer are gratefully acknowledged.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2004/5/20

Y1 - 2004/5/20

N2 - In "equation-free" multiscale computation a dynamic model is given at a fine, microscopic level; yet we believe that its coarse-grained, macroscopic dynamics can be described by closed equations involving only coarse variables. These variables are typically various low-order moments of the distributions evolved through the microscopic model. We consider the problem of integrating these unavailable equations by acting directly on kinetic Monte Carlo microscopic simulators, thus circumventing their derivation in closed form. In particular, we use projective multi-step integration to solve the coarse initial value problem forward in time as well as backward in time (under certain conditions). Macroscopic trajectories are thus traced back to unstable, source-type, and even sometimes saddle-like stationary points, even though the microscopic simulator only evolves forward in time. We also demonstrate the use of such projective integrators in a shooting boundary value problem formulation for the computation of "coarse limit cycles" of the macroscopic behavior, and the approximation of their stability through estimates of the leading "coarse Floquet multipliers".

AB - In "equation-free" multiscale computation a dynamic model is given at a fine, microscopic level; yet we believe that its coarse-grained, macroscopic dynamics can be described by closed equations involving only coarse variables. These variables are typically various low-order moments of the distributions evolved through the microscopic model. We consider the problem of integrating these unavailable equations by acting directly on kinetic Monte Carlo microscopic simulators, thus circumventing their derivation in closed form. In particular, we use projective multi-step integration to solve the coarse initial value problem forward in time as well as backward in time (under certain conditions). Macroscopic trajectories are thus traced back to unstable, source-type, and even sometimes saddle-like stationary points, even though the microscopic simulator only evolves forward in time. We also demonstrate the use of such projective integrators in a shooting boundary value problem formulation for the computation of "coarse limit cycles" of the macroscopic behavior, and the approximation of their stability through estimates of the leading "coarse Floquet multipliers".

KW - Kinetic Monte Carlo

KW - Limit cycles

KW - Projective integration

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U2 - 10.1016/j.jcp.2003.11.005

DO - 10.1016/j.jcp.2003.11.005

M3 - Article

AN - SCOPUS:2442498534

VL - 196

SP - 474

EP - 489

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 2

ER -