12,792 research outputs found
Fast simulation of large-scale growth models
We give an algorithm that computes the final state of certain growth models
without computing all intermediate states. Our technique is based on a "least
action principle" which characterizes the odometer function of the growth
process. Starting from an approximation for the odometer, we successively
correct under- and overestimates and provably arrive at the correct final
state.
Internal diffusion-limited aggregation (IDLA) is one of the models amenable
to our technique. The boundary fluctuations in IDLA were recently proved to be
at most logarithmic in the size of the growth cluster, but the constant in
front of the logarithm is still not known. As an application of our method, we
calculate the size of fluctuations over two orders of magnitude beyond previous
simulations, and use the results to estimate this constant.Comment: 27 pages, 9 figures. To appear in Random Structures & Algorithm
Baroclinic Vorticity Production in Protoplanetary Disks; Part I: Vortex Formation
The formation of vortices in protoplanetary disks is explored via
pseudo-spectral numerical simulations of an anelastic-gas model. This model is
a coupled set of equations for vorticity and temperature in two dimensions
which includes baroclinic vorticity production and radiative cooling. Vortex
formation is unambiguously shown to be caused by baroclinicity because (1)
these simulations have zero initial perturbation vorticity and a nonzero
initial temperature distribution; and (2) turning off the baroclinic term halts
vortex formation, as shown by an immediate drop in kinetic energy and
vorticity. Vortex strength increases with: larger background temperature
gradients; warmer background temperatures; larger initial temperature
perturbations; higher Reynolds number; and higher resolution. In the
simulations presented here vortices form when the background temperatures are
and vary radially as , the initial vorticity
perturbations are zero, the initial temperature perturbations are 5% of the
background, and the Reynolds number is . A sensitivity study consisting
of 74 simulations showed that as resolution and Reynolds number increase,
vortices can form with smaller initial temperature perturbations, lower
background temperatures, and smaller background temperature gradients. For the
parameter ranges of these simulations, the disk is shown to be convectively
stable by the Solberg-H{\o}iland criteria.Comment: Originally submitted to The Astrophysical Journal April 3, 2006;
resubmitted November 3, 2006; accepted Dec 5, 200
DEFROST: A New Code for Simulating Preheating after Inflation
At the end of inflation, dynamical instability can rapidly deposit the energy
of homogeneous cold inflaton into excitations of other fields. This process,
known as preheating, is rather violent, inhomogeneous and non-linear, and has
to be studied numerically. This paper presents a new code for simulating scalar
field dynamics in expanding universe written for that purpose. Compared to
available alternatives, it significantly improves both the speed and the
accuracy of calculations, and is fully instrumented for 3D visualization. We
reproduce previously published results on preheating in simple chaotic
inflation models, and further investigate non-linear dynamics of the inflaton
decay. Surprisingly, we find that the fields do not want to thermalize quite
the way one would think. Instead of directly reaching equilibrium, the
evolution appears to be stuck in a rather simple but quite inhomogeneous state.
In particular, one-point distribution function of total energy density appears
to be universal among various two-field preheating models, and is exceedingly
well described by a lognormal distribution. It is tempting to attribute this
state to scalar field turbulence.Comment: RevTeX 4.0; 16 pages, 9 figure
Fourier spectral methods for fractional-in-space reaction-diffusion equations
Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is computationally demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reactiondiffusion equations. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is show-cased by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models,together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator
Combining phase field crystal methods with a Cahn-Hilliard model for binary alloys
During phase transitions certain properties of a material change, such as
composition field and lattice-symmetry distortions. These changes are typically
coupled, and affect the microstructures that form in materials. Here, we
propose a 2D theoretical framework that couples a Cahn-Hilliard (CH) model
describing the composition field of a material system, with a phase field
crystal (PFC) model describing its underlying microscopic configurations. We
couple the two continuum models via coordinate transformation coefficients. We
introduce the transformation coefficients in the PFC method, to describe affine
lattice deformations. These transformation coefficients are modeled as
functions of the composition field. Using this coupled approach, we explore the
effects of coarse-grained lattice symmetry and distortions on a phase
transition process. In this paper, we demonstrate the working of the CH-PFC
model through three representative examples: First, we describe base cases with
hexagonal and square lattice symmetries for two composition fields. Next, we
illustrate how the CH-PFC method interpolates lattice symmetry across a diffuse
composition phase boundary. Finally, we compute a Cahn-Hilliard type of
diffusion and model the accompanying changes to lattice symmetry during a phase
transition process.Comment: 9 pages, 5 figure
Multi-scale initial conditions for cosmological simulations
We discuss a new algorithm to generate multi-scale initial conditions with
multiple levels of refinements for cosmological "zoom-in" simulations. The
method uses an adaptive convolution of Gaussian white noise with a real space
transfer function kernel together with an adaptive multi-grid Poisson solver to
generate displacements and velocities following first (1LPT) or second order
Lagrangian perturbation theory (2LPT). The new algorithm achieves RMS relative
errors of order 10^(-4) for displacements and velocities in the refinement
region and thus improves in terms of errors by about two orders of magnitude
over previous approaches. In addition, errors are localized at coarse-fine
boundaries and do not suffer from Fourier-space induced interference ringing.
An optional hybrid multi-grid and Fast Fourier Transform (FFT) based scheme is
introduced which has identical Fourier space behaviour as traditional
approaches. Using a suite of re-simulations of a galaxy cluster halo our real
space based approach is found to reproduce correlation functions, density
profiles, key halo properties and subhalo abundances with per cent level
accuracy. Finally, we generalize our approach for two-component baryon and
dark-matter simulations and demonstrate that the power spectrum evolution is in
excellent agreement with linear perturbation theory. For initial baryon density
fields, it is suggested to use the local Lagrangian approximation in order to
generate a density field for mesh based codes that is consistent with
Lagrangian perturbation theory instead of the current practice of using the
Eulerian linearly scaled densities.Comment: 22 pages, 24 figures. MNRAS in press. Updated affiliation
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