13,959 research outputs found
Moment Closure - A Brief Review
Moment closure methods appear in myriad scientific disciplines in the
modelling of complex systems. The goal is to achieve a closed form of a large,
usually even infinite, set of coupled differential (or difference) equations.
Each equation describes the evolution of one "moment", a suitable
coarse-grained quantity computable from the full state space. If the system is
too large for analytical and/or numerical methods, then one aims to reduce it
by finding a moment closure relation expressing "higher-order moments" in terms
of "lower-order moments". In this brief review, we focus on highlighting how
moment closure methods occur in different contexts. We also conjecture via a
geometric explanation why it has been difficult to rigorously justify many
moment closure approximations although they work very well in practice.Comment: short survey paper (max 20 pages) for a broad audience in
mathematics, physics, chemistry and quantitative biolog
On the Eulerian Large Eddy Simulation of disperse phase flows: an asymptotic preserving scheme for small Stokes number flows
In the present work, the Eulerian Large Eddy Simulation of dilute disperse
phase flows is investigated. By highlighting the main advantages and drawbacks
of the available approaches in the literature, a choice is made in terms of
modelling: a Fokker-Planck-like filtered kinetic equation proposed by Zaichik
et al. 2009 and a Kinetic-Based Moment Method (KBMM) based on a Gaussian
closure for the NDF proposed by Vie et al. 2014. The resulting Euler-like
system of equations is able to reproduce the dynamics of particles for small to
moderate Stokes number flows, given a LES model for the gaseous phase, and is
representative of the generic difficulties of such models. Indeed, it
encounters strong constraints in terms of numerics in the small Stokes number
limit, which can lead to a degeneracy of the accuracy of standard numerical
methods. These constraints are: 1/as the resulting sound speed is inversely
proportional to the Stokes number, it is highly CFL-constraining, and 2/the
system tends to an advection-diffusion limit equation on the number density
that has to be properly approximated by the designed scheme used for the whole
range of Stokes numbers. Then, the present work proposes a numerical scheme
that is able to handle both. Relying on the ideas introduced in a different
context by Chalons et al. 2013: a Lagrange-Projection, a relaxation formulation
and a HLLC scheme with source terms, we extend the approach to a singular flux
as well as properly handle the energy equation. The final scheme is proven to
be Asymptotic-Preserving on 1D cases comparing to either converged or
analytical solutions and can easily be extended to multidimensional
configurations, thus setting the path for realistic applications
Mean-field theory of collective motion due to velocity alignment
We introduce a system of self-propelled agents (active Brownian particles)
with velocity alignment in two spatial dimensions and derive a mean-field
theory from the microscopic dynamics via a nonlinear Fokker-Planck equation and
a moment expansion of the probability distribution function. We analyze the
stationary solutions corresponding to macroscopic collective motion with finite
center of mass velocity (ordered state) and the disordered solution with no
collective motion in the spatially homogeneous system. In particular, we
discuss the impact of two different propulsion functions governing the
individual dynamics. Our results predict a strong impact of the individual
dynamics on the mean field onset of collective motion (continuous vs
discontinuous). In addition to the macroscopic density and velocity field we
consider explicitly the dynamics of an effective temperature of the agent
system, representing a measure of velocity fluctuations around the mean
velocity. We show that the temperature decreases strongly with increasing level
of collective motion despite constant fluctuations on individual level, which
suggests that extreme caution should be taken in deducing individual behavior,
such as, state-dependent individual fluctuations from mean-field measurements
[Yates {\em et al.}, PNAS, 106 (14), 2009].Comment: corrected version, Ecological Complexity (2011) in pres
- …