4,219 research outputs found
Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations
The Navier--Stokes equations are commonly used to model and to simulate flow
phenomena. We introduce the basic equations and discuss the standard methods
for the spatial and temporal discretization. We analyse the semi-discrete
equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index
and quantify the numerical difficulties in the fully discrete schemes, that are
induced by the strangeness of the system. By analyzing the Kronecker index of
the difference-algebraic equations, that represent commonly and successfully
used time stepping schemes for the Navier--Stokes equations, we show that those
time-integration schemes factually remove the strangeness. The theoretical
considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909,
https://doi.org/10.5281/zenodo.99890
Nonlinear modes of clarinet-like musical instruments
The concept of nonlinear modes is applied in order to analyze the behavior of
a model of woodwind reed instruments. Using a modal expansion of the impedance
of the instrument, and by projecting the equation for the acoustic pressure on
the normal modes of the air column, a system of second order ordinary
differential equations is obtained. The equations are coupled through the
nonlinear relation describing the volume flow of air through the reed channel
in response to the pressure difference across the reed. The system is treated
using an amplitude-phase formulation for nonlinear modes, where the frequency
and damping functions, as well as the invariant manifolds in the phase space,
are unknowns to be determined. The formulation gives, without explicit
integration of the underlying ordinary differential equation, access to the
transient, the limit cycle, its period and stability. The process is
illustrated for a model reduced to three normal modes of the air column
Self-propelled particles with selective attraction-repulsion interaction - From microscopic dynamics to coarse-grained theories
In this work we derive and analyze coarse-grained descriptions of
self-propelled particles with selective attraction-repulsion interaction, where
individuals may respond differently to their neighbours depending on their
relative state of motion (approach versus movement away). Based on the
formulation of a nonlinear Fokker-Planck equation, we derive a kinetic
description of the system dynamics in terms of equations for the Fourier modes
of a one-particle density function. This approach allows effective numerical
investigation of the stability of possible solutions of the system. The
detailed analysis of the interaction integrals entering the equations
demonstrates that divergences at small wavelengths can appear at arbitrary
expansion orders.
Further on, we also derive a hydrodynamic theory by performing a closure at
the level of the second Fourier mode of the one-particle density function. We
show that the general form of equations is in agreement with the theory
formulated by Toner and Tu.
Finally, we compare our analytical predictions on the stability of the
disordered homogeneous solution with results of individual-based simulations.
They show good agreement for sufficiently large densities and non-negligible
short-ranged repulsion. Disagreements of numerical results and the hydrodynamic
theory for weak short-ranged repulsion reveal the existence of a previously
unknown phase of the model consisting of dense, nematically aligned filaments,
which cannot be accounted for by the present Toner and Tu type theory of polar
active matter.Comment: revised version, 37pages, 11 figure
Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials
We consider a class of fully-nonlinear Fermi-Pasta-Ulam (FPU) lattices,
consisting of a chain of particles coupled by fractional power nonlinearities
of order . This class of systems incorporates a classical Hertzian
model describing acoustic wave propagation in chains of touching beads in the
absence of precompression. We analyze the propagation of localized waves when
is close to unity. Solutions varying slowly in space and time are
searched with an appropriate scaling, and two asymptotic models of the chain of
particles are derived consistently. The first one is a logarithmic KdV
equation, and possesses linearly orbitally stable Gaussian solitary wave
solutions. The second model consists of a generalized KdV equation with
H\"older-continuous fractional power nonlinearity and admits compacton
solutions, i.e. solitary waves with compact support. When , we numerically establish the asymptotically Gaussian shape of exact FPU
solitary waves with near-sonic speed, and analytically check the pointwise
convergence of compactons towards the limiting Gaussian profile
Nonlinear analysis of spacecraft thermal models
We study the differential equations of lumped-parameter models of spacecraft
thermal control. Firstly, we consider a satellite model consisting of two
isothermal parts (nodes): an outer part that absorbs heat from the environment
as radiation of various types and radiates heat as a black-body, and an inner
part that just dissipates heat at a constant rate. The resulting system of two
nonlinear ordinary differential equations for the satellite's temperatures is
analyzed with various methods, which prove that the temperatures approach a
steady state if the heat input is constant, whereas they approach a limit cycle
if it varies periodically. Secondly, we generalize those methods to study a
many-node thermal model of a spacecraft: this model also has a stable steady
state under constant heat inputs that becomes a limit cycle if the inputs vary
periodically. Finally, we propose new numerical analyses of spacecraft thermal
models based on our results, to complement the analyses normally carried out
with commercial software packages.Comment: 29 pages, 4 figure
Gravitational collapse with non-vanishing tangential stresses II: a laboratory for cosmic censorship experiments
The general exact solution describing the dynamics of anisotropic elastic
spheres supported only by tangential stresses is reduced to a quadrature using
Ori's mass-area coordinates. This leads to the explicit construction of the
root equation governing the nature of the central singularity. Using this
equation, we formulate and motivate on physical grounds a conjecture on the
nature of this singularity. The conjecture covers a large sector of the space
of initial data; roughly speaking, it asserts that addition of a tangential
stress cannot undress a covered dust singularity. The root equation also allows
us to analyze the case of self-similar spacetimes and to get some insight on
the role of stresses in deciding the nature of the singularities in this case.Comment: 16 pages, Plain TeX forma
The Magnus expansion and some of its applications
Approximate resolution of linear systems of differential equations with
varying coefficients is a recurrent problem shared by a number of scientific
and engineering areas, ranging from Quantum Mechanics to Control Theory. When
formulated in operator or matrix form, the Magnus expansion furnishes an
elegant setting to built up approximate exponential representations of the
solution of the system. It provides a power series expansion for the
corresponding exponent and is sometimes referred to as Time-Dependent
Exponential Perturbation Theory. Every Magnus approximant corresponds in
Perturbation Theory to a partial re-summation of infinite terms with the
important additional property of preserving at any order certain symmetries of
the exact solution. The goal of this review is threefold. First, to collect a
number of developments scattered through half a century of scientific
literature on Magnus expansion. They concern the methods for the generation of
terms in the expansion, estimates of the radius of convergence of the series,
generalizations and related non-perturbative expansions. Second, to provide a
bridge with its implementation as generator of especial purpose numerical
integration methods, a field of intense activity during the last decade. Third,
to illustrate with examples the kind of results one can expect from Magnus
expansion in comparison with those from both perturbative schemes and standard
numerical integrators. We buttress this issue with a revision of the wide range
of physical applications found by Magnus expansion in the literature.Comment: Report on the Magnus expansion for differential equations and its
applications to several physical problem
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