173 research outputs found
Quasi-periodic solutions of the 2D Euler equation
We consider the two-dimensional Euler equation with periodic boundary
conditions. We construct time quasi-periodic solutions of this equation made of
localized travelling profiles with compact support propagating over a
stationary state depending on only one variable. The direction of propagation
is orthogonal to this variable, and the support is concentrated on flat strips
of the stationary state. The frequencies of the solution are given by the
locally constant velocities associated with the stationary state
A particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling
In this work, we derive particle schemes, based on micro-macro decomposition,
for linear kinetic equations in the diffusion limit. Due to the particle
approximation of the micro part, a splitting between the transport and the
collision part has to be performed, and the stiffness of both these two parts
prevent from uniform stability. To overcome this difficulty, the micro-macro
system is reformulated into a continuous PDE whose coefficients are no longer
stiff, and depend on the time step in a consistent way. This
non-stiff reformulation of the micro-macro system allows the use of standard
particle approximations for the transport part, and extends the work in [5]
where a particle approximation has been applied using a micro-macro
decomposition on kinetic equations in the fluid scaling. Beyond the so-called
asymptotic-preserving property which is satisfied by our schemes, they
significantly reduce the inherent noise of traditional particle methods, and
they have a computational cost which decreases as the system approaches the
diffusion limit
Nonlinear Geometric Optics method based multi-scale numerical schemes for highly-oscillatory transport equations
We introduce a new numerical strategy to solve a class of oscillatory
transport PDE models which is able to captureaccurately the solutions without
numerically resolving the high frequency oscillations {\em in both space and
time}.Such PDE models arise in semiclassical modeling of quantum dynamics with
band-crossings, and otherhighly oscillatory waves. Our first main idea is to
use the nonlinear geometric optics ansatz, which builds theoscillatory phase
into an independent variable. We then choose suitable initial data, based on
the Chapman-Enskog expansion, for the new model. For a scalar model, we prove
that so constructed model will have certain smoothness, and consequently, for a
first order approximation scheme we prove uniform error estimates independent
of the (possibly small) wave length. The method is extended to systems arising
from a semiclassical model for surface hopping, a non-adiabatic quantum dynamic
phenomenon. Numerous numerical examples demonstrate that the method has the
desired properties
Multiscale numerical schemes for kinetic equations in the anomalous diffusion limit
We construct numerical schemes to solve kinetic equations with anomalous
diffusion scaling. When the equilibrium is heavy-tailed or when the collision
frequency degenerates for small velocities, an appropriate scaling should be
made and the limit model is the so-called anomalous or fractional diffusion
model. Our first scheme is based on a suitable micro-macro decomposition of the
distribution function whereas our second scheme relies on a Duhamel formulation
of the kinetic equation. Both are \emph{Asymptotic Preserving} (AP): they are
consistent with the kinetic equation for all fixed value of the scaling
parameter and degenerate into a consistent scheme solving the
asymptotic model when tends to . The second scheme enjoys the
stronger property of being uniformly accurate (UA) with respect to
. The usual AP schemes known for the classical diffusion limit
cannot be directly applied to the context of anomalous diffusion scaling, since
they are not able to capture the important effects of large and small
velocities. We present numerical tests to highlight the efficiency of our
schemes
Numerical schemes for kinetic equations in the diffusion and anomalous diffusion limits. Part I: the case of heavy-tailed equilibrium
In this work, we propose some numerical schemes for linear kinetic equations
in the diffusion and anomalous diffusion limit. When the equilibrium
distribution function is a Maxwellian distribution, it is well known that for
an appropriate time scale, the small mean free path limit gives rise to a
diffusion type equation. However, when a heavy-tailed distribution is
considered, another time scale is required and the small mean free path limit
leads to a fractional anomalous diffusion equation. Our aim is to develop
numerical schemes for the original kinetic model which works for the different
regimes, without being restricted by stability conditions of standard explicit
time integrators. First, we propose some numerical schemes for the diffusion
asymptotics; then, their extension to the anomalous diffusion limit is studied.
In this case, it is crucial to capture the effect of the large velocities of
the heavy-tailed equilibrium, so that some important transformations of the
schemes derived for the diffusion asymptotics are needed. As a result, we
obtain numerical schemes which enjoy the Asymptotic Preserving property in the
anomalous diffusion limit, that is: they do not suffer from the restriction on
the time step and they degenerate towards the fractional diffusion limit when
the mean free path goes to zero. We also numerically investigate the uniform
accuracy and construct a class of numerical schemes satisfying this property.
Finally, the efficiency of the different numerical schemes is shown through
numerical experiments
Asymptotic preserving schemes for highly oscillatory kinetic equation
This work is devoted to the numerical simulation of a Vlasov-Poisson model
describing a charged particle beam under the action of a rapidly oscillating
external electric field. We construct an Asymptotic Preserving numerical scheme
for this kinetic equation in the highly oscillatory limit. This scheme enables
to simulate the problem without using any time step refinement technique.
Moreover, since our numerical method is not based on the derivation of the
simulation of asymptotic models, it works in the regime where the solution does
not oscillate rapidly, and in the highly oscillatory regime as well. Our method
is based on a "double-scale" reformulation of the initial equation, with the
introduction of an additional periodic variable
Asymptotic Preserving numerical schemes for multiscale parabolic problems
We consider a class of multiscale parabolic problems with diffusion
coefficients oscillating in space at a possibly small scale .
Numerical homogenization methods are popular for such problems, because they
capture efficiently the asymptotic behaviour as ,
without using a dramatically fine spatial discretization at the scale of the
fast oscillations. However, known such homogenization schemes are in general
not accurate for both the highly oscillatory regime
and the non oscillatory regime . In this paper, we
introduce an Asymptotic Preserving method based on an exact micro-macro
decomposition of the solution which remains consistent for both regimes.Comment: 7 pages, to appear in C. R. Acad. Sci. Paris; Ser.
Asymptotic preserving schemes for the Wigner-Poisson-BGK equations in the diffusion limit
International audienceThis work focusses on the numerical simulation of the Wigner-Poisson-BGK equation in the diffusion asymptotics. Our strategy is based on a ''micro-macro" decomposition, which leads to a system of equations that couple the macroscopic evolution (diffusion) to a microscopic kinetic contribution for the fluctuations. A semi-implicit discretization provides a numerical scheme which is stable with respect to the small parameter (mean free path) and which possesses the following properties: (i) it enjoys the asymptotic preserving property in the diffusive limit; (ii) it recovers a standard discretization of the Wigner-Poisson equation in the collisionless regime. Numerical experiments confirm the good behaviour of the numerical scheme in both regimes. The case of a spatially dependent is also investigated
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