10,769 research outputs found
Multi-scale control variate methods for uncertainty quantification in kinetic equations
Kinetic equations play a major rule in modeling large systems of interacting
particles. Uncertainties may be due to various reasons, like lack of knowledge
on the microscopic interaction details or incomplete informations at the
boundaries. These uncertainties, however, contribute to the curse of
dimensionality and the development of efficient numerical methods is a
challenge. In this paper we consider the construction of novel multi-scale
methods for such problems which, thanks to a control variate approach, are
capable to reduce the variance of standard Monte Carlo techniques
Monte Carlo computation of correlation times of independent relaxation modes at criticality
We investigate aspects of universality of Glauber critical dynamics in two
dimensions. We compute the critical exponent and numerically corroborate
its universality for three different models in the static Ising universality
class and for five independent relaxation modes. We also present evidence for
universality of amplitude ratios, which shows that, as far as dynamic behavior
is concerned, each model in a given universality class is characterized by a
single non-universal metric factor which determines the overall time scale.
This paper also discusses in detail the variational and projection methods that
are used to compute relaxation times with high accuracy
The final-parsec problem in the collisionless limit
A binary supermassive black hole loses energy via ejection of stars in a
galactic nucleus, until emission of gravitational waves becomes strong enough
to induce rapid coalescence. Evolution via the gravitational slingshot requires
that stars be continuously supplied to the binary, and it is known that in
spherical galaxies the reservoir of such stars is quickly depleted, leading to
stalling of the binary at parsec-scale separations. Recent N-body simulations
of galaxy mergers and isolated nonspherical galaxies suggest that this stalling
may not occur in less idealized systems. However, it remains unclear to what
degree these conclusions are affected by collisional relaxation, which is much
stronger in the numerical simulations than in real galaxies. In this study, we
present a novel Monte Carlo method that can efficiently deal with both
collisional and collisionless dynamics, and with galaxy models having arbitrary
shapes. We show that without relaxation, the final-parsec problem may be
overcome only in triaxial galaxies. Axisymmetry is not enough, but even a
moderate departure from axisymmetry is sufficient to keep the binary shrinking.
We find that the binary hardening rate is always substantially lower than the
maximum possible, "full-loss-cone" rate, and that it decreases with time, but
that stellar-dynamical interactions are nevertheless able to drive the binary
to coalescence on a timescale <=1 Gyr in any triaxial galaxy.Comment: 17 pages, 10 figures; matches published versio
Asymptotic-Preserving Monte Carlo methods for transport equations in the diffusive limit
We develop a new Monte Carlo method that solves hyperbolic transport
equations with stiff terms, characterized by a (small) scaling parameter. In
particular, we focus on systems which lead to a reduced problem of parabolic
type in the limit when the scaling parameter tends to zero. Classical Monte
Carlo methods suffer of severe time step limitations in these situations, due
to the fact that the characteristic speeds go to infinity in the diffusion
limit. This makes the problem a real challenge, since the scaling parameter may
differ by several orders of magnitude in the domain. To circumvent these time
step limitations, we construct a new, asymptotic-preserving Monte Carlo method
that is stable independently of the scaling parameter and degenerates to a
standard probabilistic approach for solving the limiting equation in the
diffusion limit. The method uses an implicit time discretization to formulate a
modified equation in which the characteristic speeds do not grow indefinitely
when the scaling factor tends to zero. The resulting modified equation can
readily be discretized by a Monte Carlo scheme, in which the particles combine
a finite propagation speed with a time-step dependent diffusion term. We show
the performance of the method by comparing it with standard (deterministic)
approaches in the literature
Towards an ultra efficient kinetic scheme. Part I: basics on the BGK equation
In this paper we present a new ultra efficient numerical method for solving
kinetic equations. In this preliminary work, we present the scheme in the case
of the BGK relaxation operator. The scheme, being based on a splitting
technique between transport and collision, can be easily extended to other
collisional operators as the Boltzmann collision integral or to other kinetic
equations such as the Vlasov equation. The key idea, on which the method
relies, is to solve the collision part on a grid and then to solve exactly the
transport linear part by following the characteristics backward in time. The
main difference between the method proposed and semi-Lagrangian methods is that
here we do not need to reconstruct the distribution function at each time step.
This allows to tremendously reduce the computational cost of the method and it
permits for the first time, to the author's knowledge, to compute solutions of
full six dimensional kinetic equations on a single processor laptop machine.
Numerical examples, up to the full three dimensional case, are presented which
validate the method and assess its efficiency in 1D, 2D and 3D
Hyperbolic character of the angular moment equations of radiative transfer and numerical methods
We study the mathematical character of the angular moment equations of
radiative transfer in spherical symmetry and conclude that the system is
hyperbolic for general forms of the closure relation found in the literature.
Hyperbolicity and causality preservation lead to mathematical conditions
allowing to establish a useful characterization of the closure relations. We
apply numerical methods specifically designed to solve hyperbolic systems of
conservation laws (the so-called Godunov-type methods), to calculate numerical
solutions of the radiation transport equations in a static background. The
feasibility of the method in any kind of regime, from diffusion to
free-streaming, is demonstrated by a number of numerical tests and the effect
of the choice of the closure relation on the results is discussed.Comment: 37 pags, 12 figures, accepted for publication in MNRA
Linear response conductance and magneto-resistance of ferromagnetic single-electron transistors
The current through ferromagnetic single-electron transistors (SET's) is
considered. Using path integrals the linear response conductance is formulated
as a function of the tunnel conductance vs. quantum conductance and the
temperature vs. Coulomb charging energy. The magneto-resistance of
ferromagnet-normal metal-ferromagnet (F-N-F) SET's is almost independent of the
Coulomb charging energy and is only reduced when the transport dwell time is
longer than the spin-flip relaxation time. In all-ferromagnetic (F-F-F) SET's
with negligible spin-flip relaxation time the magneto-resistance is calculated
analytically at high temperatures and numerically at low temperatures. The
F-F-F magneto-resistance is enhanced by higher order tunneling processes at low
temperatures in the 'off' state when the induced charges vanishes. In contrast,
in the 'on' state near resonance the magneto-resistance ratio is a
non-monotonic function of the inverse temperature.Comment: 10 pages, 6 figures. accepted for publication in Phys. Rev.
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