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Calculating effective diffusiveness in the limit of vanishing molecular diffusion

Calculating effective diffusiveness in the limit of vanishing molecular diffusion
Calculating effective diffusiveness in the limit of vanishing molecular diffusion
In this paper we study the problem of the numerical calculation (by Monte Carlo methods) of the effective diffusivity for a particle moving in a periodic divergent-free velocity field, in the limit of vanishing molecular diffusion. In this limit traditional numerical methods typically fail, since they do not represent accurately the geometry of the underlying deterministic dynamics. We propose a stochastic splitting method that takes into account the volume-preserving property of the equations of motion in the absence of noise, and when inertial effects can be neglected. An extension of the method is then proposed for the cases where the noise has a non-trivial time-correlation structure and when inertial effects cannot be neglected. The method of modified equations is used to explain failings of Euler-based methods. The new stochastic geometric integrators are shown to outperform standard Euler-based integrators. Various asymptotic limits of physical interest are investigated by means of numerical experiments, using the new integrators.

0021-9991
1030-1055
Pavliotis, G.A.
2041971d-67c4-4cb0-88c9-c7b2443f59d7
Stuart, A.M.
0b611978-6d94-488c-b92b-09cfc1a2ceb2
Zygalakis, K.C.
a330d719-2ccb-49bd-8cd8-d06b1e6daca6
Pavliotis, G.A.
2041971d-67c4-4cb0-88c9-c7b2443f59d7
Stuart, A.M.
0b611978-6d94-488c-b92b-09cfc1a2ceb2
Zygalakis, K.C.
a330d719-2ccb-49bd-8cd8-d06b1e6daca6

Pavliotis, G.A., Stuart, A.M. and Zygalakis, K.C. (2009) Calculating effective diffusiveness in the limit of vanishing molecular diffusion. Journal of Computational Physics, 228 (4), 1030-1055. (doi:10.1016/j.jcp.2008.10.014).

Record type: Article

Abstract

In this paper we study the problem of the numerical calculation (by Monte Carlo methods) of the effective diffusivity for a particle moving in a periodic divergent-free velocity field, in the limit of vanishing molecular diffusion. In this limit traditional numerical methods typically fail, since they do not represent accurately the geometry of the underlying deterministic dynamics. We propose a stochastic splitting method that takes into account the volume-preserving property of the equations of motion in the absence of noise, and when inertial effects can be neglected. An extension of the method is then proposed for the cases where the noise has a non-trivial time-correlation structure and when inertial effects cannot be neglected. The method of modified equations is used to explain failings of Euler-based methods. The new stochastic geometric integrators are shown to outperform standard Euler-based integrators. Various asymptotic limits of physical interest are investigated by means of numerical experiments, using the new integrators.

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Published date: March 2009
Organisations: Applied Mathematics

Identifiers

Local EPrints ID: 340551
URI: http://eprints.soton.ac.uk/id/eprint/340551
ISSN: 0021-9991
PURE UUID: 365623ab-c670-4898-906a-258e54275f29

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Date deposited: 26 Jun 2012 10:30
Last modified: 14 Mar 2024 11:26

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Contributors

Author: G.A. Pavliotis
Author: A.M. Stuart
Author: K.C. Zygalakis

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