58,348 research outputs found
Basic Concepts Underlying Singular Perturbation Techniques
In many singular perturbation problems multiple scales are used. For instance, one may use both the coordinate x and the coordinate x^* = ε^(-1)x. In a secular-type problem x and x^* are used simultaneously. This paper discusses layer-type problems in which x^* is used in a thin layer and x outside this layer. Assume one seeks approximations to a function f(x,ε), uniformly valid to some order in ε for x in a closed interval D. In layer-type problems one uses (at least) two expansions (called inner and outer) neither of which is uniformly valid but whose domains of validity together cover the interval D. To define "domain of validity" one needs to consider intervals whose endpoints depend on epsilon. In the construction of the inner and outer expansions, constants and functions of e occur which are determined by comparison of the two expansions "matching." The comparison is possible only in the domain of overlap of their regions of validity. Once overlap is established, matching is easily carried out. Heuristic ideas for determining domains of validity of approximations by a study of the corresponding equations are illustrated with the aid of model equations. It is shown that formally small terms in an equation may have large integrated effects. The study of this is of central importance for understanding layer-type problems. It is emphasized that considering the expansions as the result of applying limit processes can lead to serious errors and, in any case, hides the nature of the expansions
Formation of clumps and patches in self-aggregation of finite size particles
New model equations are derived for dynamics of self-aggregation of
finite-size particles. Differences from standard Debye-Huckel and Keller-Segel
models are: a) the mobility of particles depends on the locally-averaged
particle density and b) linear diffusion acts on that locally-averaged particle
density. The cases both with and without diffusion are considered here.
Surprisingly, these simple modifications of standard models allow progress in
the analytical description of evolution as well as the complete analysis of
stationary states. When remains positive, the evolution of collapsed
states in our model reduces exactly to finite-dimensional dynamics of
interacting particle clumps. Simulations show these collapsed (clumped) states
emerging from smooth initial conditions, even in one spatial dimension. If
vanishes for some averaged density, the evolution leads to spontaneous
formation of \emph{jammed patches} (weak solution with density having compact
support). Simulations confirm that a combination of these patches forms the
final state for the system.Comment: 38 pages, 8 figures; submitted to Physica
Approximate solution to a hybrid model with stochastic volatility: a singular-perturbation strategy
We study a hybrid model of Schobel-Zhu-Hull-White-type from a singular-perturbation-analysis perspective. The merit of the paper is twofold: On one hand, we find boundary conditions for the deterministic non-linear degenerate parabolic partial differential equation for the evolution of the stock price. On the other hand, we combine two-scales regular- and singular-perturbation techniques to find an approximate solution to the pricing PDE. The aim is to produce an expression that can be evaluated numerically very fast
A fluctuating boundary integral method for Brownian suspensions
We present a fluctuating boundary integral method (FBIM) for overdamped
Brownian Dynamics (BD) of two-dimensional periodic suspensions of rigid
particles of complex shape immersed in a Stokes fluid. We develop a novel
approach for generating Brownian displacements that arise in response to the
thermal fluctuations in the fluid. Our approach relies on a first-kind boundary
integral formulation of a mobility problem in which a random surface velocity
is prescribed on the particle surface, with zero mean and covariance
proportional to the Green's function for Stokes flow (Stokeslet). This approach
yields an algorithm that scales linearly in the number of particles for both
deterministic and stochastic dynamics, handles particles of complex shape,
achieves high order of accuracy, and can be generalized to three dimensions and
other boundary conditions. We show that Brownian displacements generated by our
method obey the discrete fluctuation-dissipation balance relation (DFDB). Based
on a recently-developed Positively Split Ewald method [A. M. Fiore, F. Balboa
Usabiaga, A. Donev and J. W. Swan, J. Chem. Phys., 146, 124116, 2017],
near-field contributions to the Brownian displacements are efficiently
approximated by iterative methods in real space, while far-field contributions
are rapidly generated by fast Fourier-space methods based on fluctuating
hydrodynamics. FBIM provides the key ingredient for time integration of the
overdamped Langevin equations for Brownian suspensions of rigid particles. We
demonstrate that FBIM obeys DFDB by performing equilibrium BD simulations of
suspensions of starfish-shaped bodies using a random finite difference temporal
integrator.Comment: Submitted to J. Comp. Phy
Propagation and Structure of Planar Streamer Fronts
Streamers often constitute the first stage of dielectric breakdown in strong
electric fields: a nonlinear ionization wave transforms a non-ionized medium
into a weakly ionized nonequilibrium plasma. New understanding of this old
phenomenon can be gained through modern concepts of (interfacial) pattern
formation. As a first step towards an effective interface description, we
determine the front width, solve the selection problem for planar fronts and
calculate their properties. Our results are in good agreement with many
features of recent three-dimensional numerical simulations.
In the present long paper, you find the physics of the model and the
interfacial approach further explained. As a first ingredient of this approach,
we here analyze planar fronts, their profile and velocity. We encounter a
selection problem, recall some knowledge about such problems and apply it to
planar streamer fronts. We make analytical predictions on the selected front
profile and velocity and confirm them numerically.
(abbreviated abstract)Comment: 23 pages, revtex, 14 ps file
A model differential equation for turbulence
A phenomenological turbulence model in which the energy spectrum obeys a
nonlinear diffusion equation is presented. This equation respects the scaling
properties of the original Navier-Stokes equations and it has the Kolmogorov
-5/3 cascade and the thermodynamic equilibrium spectra as exact steady state
solutions. The general steady state in this model contains a nonlinear mixture
of the constant-flux and thermodynamic components. Such "warm cascade"
solutions describe the bottleneck phenomenon of spectrum stagnation near the
dissipative scale. Self-similar solutions describing a finite-time formation of
steady cascades are analysed and found to exhibit nontrivial scaling behaviour.Comment: April 10 2003 Updated April 22 2003, 9 pages revtex4, 9 figures Added
some figures, additional references and corrected typo
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