61 research outputs found
Analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit
We present a mathematical analysis of the asymptotic preserving scheme
proposed in [M. Lemou and L. Mieussens, SIAM J. Sci. Comput., 31, pp. 334-368,
2008] for linear transport equations in kinetic and diffusive regimes. We prove
that the scheme is uniformly stable and accurate with respect to the mean free
path of the particles. This property is satisfied under an explicitly given CFL
condition. This condition tends to a parabolic CFL condition for small mean
free paths, and is close to a convection CFL condition for large mean free
paths. Ou r analysis is based on very simple energy estimates
Numerical schemes of diffusion asymptotics and moment closures for kinetic equations
We investigate different models that are intended to describe the small mean free path regime of a kinetic equation, a particular attention being paid to the moment closure by entropy minimization. We introduce a specific asymptotic-induced numerical strategy which is able to treat the stiff terms of the asymptotic diffusive regime. We evaluate on numerics the performances of the method and the abilities of the reduced models to capture the main features of the full kinetic equation
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
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
Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit
We consider Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolic
systems with stiff relaxation in the so-called diffusion limit. In such regime
the system relaxes towards a convection-diffusion equation. The first objective
of the paper is to show that traditional partitioned IMEX R-K schemes will
relax to an explicit scheme for the limit equation with no need of modification
of the original system. Of course the explicit scheme obtained in the limit
suffers from the classical parabolic stability restriction on the time step.
The main goal of the paper is to present an approach, based on IMEX R-K
schemes, that in the diffusion limit relaxes to an IMEX R-K scheme for the
convection-diffusion equation, in which the diffusion is treated implicitly.
This is achieved by an original reformulation of the problem, and subsequent
application of IMEX R-K schemes to it. An analysis on such schemes to the
reformulated problem shows that the schemes reduce to IMEX R-K schemes for the
limit equation, under the same conditions derived for hyperbolic relaxation.
Several numerical examples including neutron transport equations confirm the
theoretical analysis
A scheme for radiation pressure and photon diffusion with the M1 closure in RAMSES-RT
We describe and test an updated version of radiation-hydrodynamics (RHD) in
the RAMSES code, that includes three new features: i) radiation pressure on
gas, ii) accurate treatment of radiation diffusion in an unresolved optically
thick medium, and iii) relativistic corrections that account for Doppler
effects and work done by the radiation to first order in v/c. We validate the
implementation in a series of tests, which include a morphological assessment
of the M1 closure for the Eddington tensor in an astronomically relevant
setting, dust absorption in a optically semi-thick medium, direct pressure on
gas from ionising radiation, convergence of our radiation diffusion scheme
towards resolved optical depths, correct diffusion of a radiation flash and a
constant luminosity radiation, and finally, an experiment from Davis et al. of
the competition between gravity and radiation pressure in a dusty atmosphere,
and the formation of radiative Rayleigh-Taylor instabilities. With the new
features, RAMSES-RT can be used for state-of-the-art simulations of radiation
feedback from first principles, on galactic and cosmological scales, including
not only direct radiation pressure from ionising photons, but also indirect
pressure via dust from multi-scattered IR photons reprocessed from
higher-energy radiation, both in the optically thin and thick limits.Comment: 25 pages, 13 figures, accepted for publication in MNRAS. Revised to
match published versio
An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits.
International audienceIn this work, we extend the micro-macro decomposition based numerical schemes developed in \cite{benoune} to the collisional Vlasov-Poisson model in the diffusion and high-field asymptotics. In doing so, we first write the Vlasov-Poisson model as a system that couples the macroscopic (equilibrium) part with the remainder part. A suitable discretization of this micro-macro model enables to derive an asymptotic preserving scheme in the diffusion and high-field asymptotics. In addition, two main improvements are presented: On the one hand a self-consistent electric field is introduced, which induces a specific discretization in the velocity direction, and represents a wide range of applications in plasma physics. On the other hand, as suggested in \cite{noteL}, we introduce a suitable reformulation of the micro-macro scheme which leads to an asymptotic preserving property with the following property: It degenerates into an implicit scheme for the diffusion limit model when , which makes it free from the usual diffusion constraint in all regimes. Numerical examples are used to demonstrate the efficiency and the applicability of the schemes for both regimes
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