3,117 research outputs found
Convergence of a Finite Volume Scheme for a Corrosion Model
In this paper, we study the numerical approximation of a system of partial
dif-ferential equations describing the corrosion of an iron based alloy in a
nuclear waste repository. In particular, we are interested in the convergence
of a numerical scheme consisting in an implicit Euler scheme in time and a
Scharfetter-Gummel finite volume scheme in space
Self-Similar Evolution of Cosmic-Ray-Modified Quasi-Parallel Plane Shocks
Using an improved version of the previously introduced CRASH (Cosmic Ray
Acceleration SHock) code, we have calculated the time evolution of cosmic-ray
(CR) modified quasi-parallel plane shocks for Bohm-like diffusion, including
self-consistent models of Alfven wave drift and dissipation, along with thermal
leakage injection of CRs. The new simulations follow evolution of the CR
distribution to much higher energies than our previous study, providing a
better examination of evolutionary and asymptotic behaviors. The postshock CR
pressure becomes constant after quick initial adjustment, since the evolution
of the CR partial pressure expressed in terms of a momentum similarity variable
is self-similar. The shock precursor, which scales as the diffusion length of
the highest energy CRs, subsequently broadens approximately linearly with time,
independent of diffusion model, so long as CRs continue to be accelerated to
ever-higher energies. This means the nonlinear shock structure can be described
approximately in terms of the similarity variable, x/(u_s t), where u_s is the
shock speed once the postshock pressure reaches an approximate time asymptotic
state. As before, the shock Mach number is the key parameter determining the
evolution and the CR acceleration efficiency, although finite Alfven wave drift
and wave energy dissipation in the shock precursor reduce the effective
velocity change experienced by CRs, so reduce acceleration efficiency
noticeably, thus, providing a second important parameter at low and moderate
Mach numbers.Comment: 29 pages, 8 figure
Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit
This paper analyzes various schemes for the Euler-Poisson-Boltzmann (EPB)
model of plasma physics. This model consists of the pressureless gas dynamics
equations coupled with the Poisson equation and where the Boltzmann relation
relates the potential to the electron density. If the quasi-neutral assumption
is made, the Poisson equation is replaced by the constraint of zero local
charge and the model reduces to the Isothermal Compressible Euler (ICE) model.
We compare a numerical strategy based on the EPB model to a strategy using a
reformulation (called REPB formulation). The REPB scheme captures the
quasi-neutral limit more accurately
A Finite-Volume Scheme for a Spinorial Matrix Drift-Diffusion Model for Semiconductors
An implicit Euler finite-volume scheme for a spinorial matrix drift-diffusion
model for semiconductors is analyzed. The model consists of strongly coupled
parabolic equations for the electron density matrix or, alternatively, of
weakly coupled equations for the charge and spin-vector densities, coupled to
the Poisson equation for the elec-tric potential. The equations are solved in a
bounded domain with mixed Dirichlet-Neumann boundary conditions. The charge and
spin-vector fluxes are approximated by a Scharfetter-Gummel discretization. The
main features of the numerical scheme are the preservation of positivity and L
bounds and the dissipation of the discrete free energy. The existence
of a bounded discrete solution and the monotonicity of the discrete free energy
are proved. For undoped semiconductor materials, the numerical scheme is
uncon-ditionally stable. The fundamental ideas are reformulations using spin-up
and spin-down densities and certain projections of the spin-vector density,
free energy estimates, and a discrete Moser iteration. Furthermore, numerical
simulations of a simple ferromagnetic-layer field-effect transistor in two
space dimensions are presented
Kinetic theory of jet dynamics in the stochastic barotropic and 2D Navier-Stokes equations
We discuss the dynamics of zonal (or unidirectional) jets for barotropic
flows forced by Gaussian stochastic fields with white in time correlation
functions. This problem contains the stochastic dynamics of 2D Navier-Stokes
equation as a special case. We consider the limit of weak forces and
dissipation, when there is a time scale separation between the inertial time
scale (fast) and the spin-up or spin-down time (large) needed to reach an
average energy balance. In this limit, we show that an adiabatic reduction (or
stochastic averaging) of the dynamics can be performed. We then obtain a
kinetic equation that describes the slow evolution of zonal jets over a very
long time scale, where the effect of non-zonal turbulence has been integrated
out. The main theoretical difficulty, achieved in this work, is to analyze the
stationary distribution of a Lyapunov equation that describes quasi-Gaussian
fluctuations around each zonal jet, in the inertial limit. This is necessary to
prove that there is no ultraviolet divergence at leading order in such a way
that the asymptotic expansion is self-consistent. We obtain at leading order a
Fokker--Planck equation, associated to a stochastic kinetic equation, that
describes the slow jet dynamics. Its deterministic part is related to well
known phenomenological theories (for instance Stochastic Structural Stability
Theory) and to quasi-linear approximations, whereas the stochastic part allows
to go beyond the computation of the most probable zonal jet. We argue that the
effect of the stochastic part may be of huge importance when, as for instance
in the proximity of phase transitions, more than one attractor of the dynamics
is present
Well posedness and Maximum Entropy Approximation for the Dynamics of Quantitative Traits
We study the Fokker-Planck equation derived in the large system limit of the
Markovian process describing the dynamics of quantitative traits. The
Fokker-Planck equation is posed on a bounded domain and its transport and
diffusion coefficients vanish on the domain's boundary. We first argue that,
despite this degeneracy, the standard no-flux boundary condition is valid. We
derive the weak formulation of the problem and prove the existence and
uniqueness of its solutions by constructing the corresponding contraction
semigroup on a suitable function space. Then, we prove that for the parameter
regime with high enough mutation rate the problem exhibits a positive spectral
gap, which implies exponential convergence to equilibrium.
Next, we provide a simple derivation of the so-called Dynamic Maximum Entropy
(DynMaxEnt) method for approximation of moments of the Fokker-Planck solution,
which can be interpreted as a nonlinear Galerkin approximation. The limited
applicability of the DynMaxEnt method inspires us to introduce its modified
version that is valid for the whole range of admissible parameters. Finally, we
present several numerical experiments to demonstrate the performance of both
the original and modified DynMaxEnt methods. We observe that in the parameter
regimes where both methods are valid, the modified one exhibits slightly better
approximation properties compared to the original one.Comment: 28 pages, 4 tables, 5 figure
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
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