1,070 research outputs found
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
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
An Asymptotic Preserving Scheme for the Euler equations in a strong magnetic field
This paper is concerned with the numerical approximation of the isothermal
Euler equations for charged particles subject to the Lorentz force. When the
magnetic field is large, the so-called drift-fluid approximation is obtained.
In this limit, the parallel motion relative to the magnetic field direction
splits from perpendicular motion and is given implicitly by the constraint of
zero total force along the magnetic field lines. In this paper, we provide a
well-posed elliptic equation for the parallel velocity which in turn allows us
to construct an Asymptotic-Preserving (AP) scheme for the Euler-Lorentz system.
This scheme gives rise to both a consistent approximation of the Euler-Lorentz
model when epsilon is finite and a consistent approximation of the drift limit
when epsilon tends to 0. Above all, it does not require any constraint on the
space and time steps related to the small value of epsilon. Numerical results
are presented, which confirm the AP character of the scheme and its Asymptotic
Stability
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 Multiscale Kinetic-Fluid Solver with Dynamic Localization of Kinetic Effects
This paper collects the efforts done in our previous works [P. Degond, S.
Jin, L. Mieussens, A Smooth Transition Between Kinetic and Hydrodynamic
Equations, J. Comp. Phys., 209 (2005) 665--694.],[P.Degond, G. Dimarco, L.
Mieussens, A Moving Interface Method for Dynamic Kinetic-fluid Coupling, J.
Comp. Phys., Vol. 227, pp. 1176-1208, (2007).],[P. Degond, J.G. Liu, L.
Mieussens, Macroscopic Fluid Model with Localized Kinetic Upscaling Effects,
SIAM Multi. Model. Sim. 5(3), 940--979 (2006)] to build a robust multiscale
kinetic-fluid solver. Our scope is to efficiently solve fluid dynamic problems
which present non equilibrium localized regions that can move, merge, appear or
disappear in time. The main ingredients of the present work are the followings
ones: a fluid model is solved in the whole domain together with a localized
kinetic upscaling term that corrects the fluid model wherever it is necessary;
this multiscale description of the flow is obtained by using a micro-macro
decomposition of the distribution function [P. Degond, J.G. Liu, L. Mieussens,
Macroscopic Fluid Model with Localized Kinetic Upscaling Effects, SIAM Multi.
Model. Sim. 5(3), 940--979 (2006)]; the dynamic transition between fluid and
kinetic descriptions is obtained by using a time and space dependent transition
function; to efficiently define the breakdown conditions of fluid models we
propose a new criterion based on the distribution function itself. Several
numerical examples are presented to validate the method and measure its
computational efficiency.Comment: 34 page
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
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 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
On massless electron limit for a multispecies kinetic system with external magnetic field
We consider a three-dimensional kinetic model for a two species plasma
consisting of electrons and ions confined by an external nonconstant magnetic
field. Then we derive a kinetic-fluid model when the mass ratio tends
to zero. Each species initially obeys a Vlasov-type equation and the
electrostatic coupling follows from a Poisson equation. In our modeling, ions
are assumed non-collisional while a Fokker-Planck collision operator is taken
into account in the electron equation. As the mass ratio tends to zero we show
convergence to a new system where the macroscopic electron density satisfies an
anisotropic drift-diffusion equation. To achieve this task, we overcome some
specific technical issues of our model such as the strong effect of the
magnetic field on electrons and the lack of regularity at the limit. With
methods usually adapted to diffusion limit of collisional kinetic equations and
including renormalized solutions, relative entropy dissipation and velocity
averages, we establish the rigorous derivation of the limit model
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