384 research outputs found
Fluid Simulations with Localized Boltzmann Upscaling by Direct Simulation Monte-Carlo
In the present work, we present a novel numerical algorithm to couple the
Direct Simulation Monte Carlo method (DSMC) for the solution of the Boltzmann
equation with a finite volume like method for the solution of the Euler
equations. Recently we presented in [14],[16],[17] different methodologies
which permit to solve fluid dynamics problems with localized regions of
departure from thermodynamical equilibrium. The methods rely on the
introduction of buffer zones which realize a smooth transition between the
kinetic and the fluid regions. In this paper we extend the idea of buffer zones
and dynamic coupling to the case of the Monte Carlo methods. To facilitate the
coupling and avoid the onset of spurious oscillations in the fluid regions
which are consequences of the coupling with a stochastic numerical scheme, we
use a new technique which permits to reduce the variance of the particle
methods [11]. In addition, the use of this method permits to obtain estimations
of the breakdowns of the fluid models less affected by fluctuations and
consequently to reduce the kinetic regions and optimize the coupling. In the
last part of the paper several numerical examples are presented to validate the
method and measure its computational performances
A multi-layer model for self-propelled disks interacting through alignment and volume exclusion
We present an individual-based model describing disk-like self-propelled
particles moving inside parallel planes. The disk directions of motion follow
alignment rules inside each layer. Additionally, the disks are subject to
interactions with those of the neighboring layers arising from volume exclusion
constraints. These interactions affect the disk inclinations with respect to
the plane of motion. We formally de-rive a macroscopic model composed of planar
Self-Organized Hydrodynamic (SOH) models describing the transport of mass and
evolution of mean direction of motion of the disks in each plane, supplemented
with transport equations for the mean disk inclination. These planar models are
coupled due to the interactions with the neighboring planes. Numerical
comparisons between the individual-based and macroscopic models are carried
out. These models could be applicable, for instance, to describe sperm-cell
collective dynamics
Topological interactions in a Boltzmann-type framework
We consider a finite number of particles characterised by their positions and
velocities. At random times a randomly chosen particle, the follower, adopts
the velocity of another particle, the leader. The follower chooses its leader
according to the proximity rank of the latter with respect to the former. We
study the limit of a system size going to infinity and, under the assumption of
propagation of chaos, show that the limit equation is akin to the Boltzmann
equation. However, it exhibits a spatial non-locality instead of the classical
non-locality in velocity space. This result relies on the approximation
properties of Bernstein polynomials
A macroscopic model for a system of swarming agents using curvature control
In this paper, we study the macroscopic limit of a new model of collective
displacement. The model, called PTWA, is a combination of the Vicsek alignment
model and the Persistent Turning Walker (PTW) model of motion by curvature
control. The PTW model was designed to fit measured trajectories of individual
fish. The PTWA model (Persistent Turning Walker with Alignment) describes the
displacements of agents which modify their curvature in order to align with
their neighbors. The derivation of its macroscopic limit uses the non-classical
notion of generalized collisional invariant. The macroscopic limit of the PTWA
model involves two physical quantities, the density and the mean velocity of
individuals. It is a system of hyperbolic type but is non-conservative due to a
geometric constraint on the velocity. This system has the same form as the
macroscopic limit of the Vicsek model (the 'Vicsek hydrodynamics') but for the
expression of the model coefficients. The numerical computations show that the
numerical values of the coefficients are very close. The 'Vicsek Hydrodynamic
model' appears in this way as a more generic macroscopic model of swarming
behavior as originally anticipated
All speed scheme for the low mach number limit of the Isentropic Euler equation
An all speed scheme for the Isentropic Euler equation is presented in this
paper. When the Mach number tends to zero, the compressible Euler equation
converges to its incompressible counterpart, in which the density becomes a
constant. Increasing approximation errors and severe stability constraints are
the main difficulty in the low Mach regime. The key idea of our all speed
scheme is the special semi-implicit time discretization, in which the low Mach
number stiff term is divided into two parts, one being treated explicitly and
the other one implicitly. Moreover, the flux of the density equation is also
treated implicitly and an elliptic type equation is derived to obtain the
density. In this way, the correct limit can be captured without requesting the
mesh size and time step to be smaller than the Mach number. Compared with
previous semi-implicit methods, nonphysical oscillations can be suppressed. We
develop this semi-implicit time discretization in the framework of a first
order local Lax-Friedrich (LLF) scheme and numerical tests are displayed to
demonstrate its performances
Hydrodynamics of self-alignment interactions with precession and derivation of the Landau-Lifschitz-Gilbert equation
We consider a kinetic model of self-propelled particles with alignment
interaction and with precession about the alignment direction. We derive a
hydrodynamic system for the local density and velocity orientation of the
particles. The system consists of the conservative equation for the local
density and a non-conservative equation for the orientation. First, we assume
that the alignment interaction is purely local and derive a first order system.
However, we show that this system may lose its hyperbolicity. Under the
assumption of weakly non-local interaction, we derive diffusive corrections to
the first order system which lead to the combination of a heat flow of the
harmonic map and Landau-Lifschitz-Gilbert dynamics. In the particular case of
zero self-propelling speed, the resulting model reduces to the phenomenological
Landau-Lifschitz-Gilbert equations. Therefore the present theory provides a
kinetic formulation of classical micromagnetization models and spin dynamics
Degenerate anisotropic elliptic problems and magnetized plasma simulations
This paper is devoted to the numerical approximation of a degenerate
anisotropic elliptic problem. The numerical method is designed for arbitrary
space-dependent anisotropy directions and does not require any specially
adapted coordinate system. It is also designed to be equally accurate in the
strongly and the mildly anisotropic cases. The method is applied to the
Euler-Lorentz system, in the drift-fluid limit. This system provides a model
for magnetized plasmas
Local stability of perfect alignment for a spatially homogeneous kinetic model
We prove the nonlinear local stability of Dirac masses for a kinetic model of
alignment of particles on the unit sphere, each point of the unit sphere
representing a direction. A population concentrated in a Dirac mass then
corresponds to the global alignment of all individuals. The main difficulty of
this model is the lack of conserved quantities and the absence of an energy
that would decrease for any initial condition. We overcome this difficulty
thanks to a functional which is decreasing in time in a neighborhood of any
Dirac mass (in the sense of the Wasserstein distance). The results are then
extended to the case where the unit sphere is replaced by a general Riemannian
manifold
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