2,391 research outputs found
On the resolvent condition in the Kreiss matrix theorem
The Kreiss Matrix Theorem asserts the uniform equivalence over all N x N matrices of power boundedness and a certain resolvent estimate. It is shown that the ratio of the constants in these two conditions grows linearly with N, and the optimal proportionality factor is obtained up to a factor of 2. Analogous results are also given for the related problem involving matrix exponentials. The proofs make use of a lemma that may be of independent interest, which bounds the arch length of the image of a circle in the complex plane under a rational function
Fourier analysis of the SOR iteration
The SOR iteration for solving linear systems of equations depends upon an overrelaxation factor omega. It is shown that for the standard model problem of Poisson's equation on a rectangle, the optimal omega and corresponding convergence rate can be rigorously obtained by Fourier analysis. The trick is to tilt the space-time grid so that the SOR stencil becomes symmetrical. The tilted grid also gives insight into the relation between convergence rates of several variants
The GeoClaw software for depth-averaged flows with adaptive refinement
Many geophysical flow or wave propagation problems can be modeled with
two-dimensional depth-averaged equations, of which the shallow water equations
are the simplest example. We describe the GeoClaw software that has been
designed to solve problems of this nature, consisting of open source Fortran
programs together with Python tools for the user interface and flow
visualization. This software uses high-resolution shock-capturing finite volume
methods on logically rectangular grids, including latitude--longitude grids on
the sphere. Dry states are handled automatically to model inundation. The code
incorporates adaptive mesh refinement to allow the efficient solution of
large-scale geophysical problems. Examples are given illustrating its use for
modeling tsunamis, dam break problems, and storm surge. Documentation and
download information is available at www.clawpack.org/geoclawComment: 18 pages, 11 figures, Animations and source code for some examples at
http://www.clawpack.org/links/awr10 Significantly modified from original
posting to incorporate suggestions of referee
Shear-Improved Smagorinsky Model for Large-Eddy Simulation of Wall-Bounded Turbulent Flows
A shear-improved Smagorinsky model is introduced based on recent results
concerning shear effects in wall-bounded turbulence by Toschi et al. (2000).
The Smagorinsky eddy-viscosity is modified subtracting the magnitude of the
mean shear from the magnitude of the instantaneous resolved strain-rate tensor.
This subgrid-scale model is tested in large-eddy simulations of plane-channel
flows at two different Reynolds numbers. First comparisons with the dynamic
Smagorinsky model and direct numerical simulations, including mean velocity,
turbulent kinetic energy and Reynolds stress profiles, are shown to be
extremely satisfactory. The proposed model, in addition of being physically
sound, has a low computational cost and possesses a high potentiality of
generalization to more complex non-homogeneous turbulent flows.Comment: 10 pages, 6 figures, added some reference
On the scaling of entropy viscosity in high order methods
In this work, we outline the entropy viscosity method and discuss how the
choice of scaling influences the size of viscosity for a simple shock problem.
We present examples to illustrate the performance of the entropy viscosity
method under two distinct scalings
Type II critical phenomena of neutron star collapse
We investigate spherically-symmetric, general relativistic systems of
collapsing perfect fluid distributions. We consider neutron star models that
are driven to collapse by the addition of an initially "in-going" velocity
profile to the nominally static star solution. The neutron star models we use
are Tolman-Oppenheimer-Volkoff solutions with an initially isentropic,
gamma-law equation of state. The initial values of 1) the amplitude of the
velocity profile, and 2) the central density of the star, span a parameter
space, and we focus only on that region that gives rise to Type II critical
behavior, wherein black holes of arbitrarily small mass can be formed. In
contrast to previously published work, we find that--for a specific value of
the adiabatic index (Gamma = 2)--the observed Type II critical solution has
approximately the same scaling exponent as that calculated for an
ultrarelativistic fluid of the same index. Further, we find that the critical
solution computed using the ideal-gas equations of state asymptotes to the
ultrarelativistic critical solution.Comment: 24 pages, 22 figures, RevTeX 4, submitted to Phys. Rev.
Numerical Simulation of the Hydrodynamical Combustion to Strange Quark Matter
We present results from a numerical solution to the burning of neutron matter
inside a cold neutron star into stable (u,d,s) quark matter. Our method solves
hydrodynamical flow equations in 1D with neutrino emission from weak
equilibrating reactions, and strange quark diffusion across the burning front.
We also include entropy change due to heat released in forming the stable quark
phase. Our numerical results suggest burning front laminar speeds of 0.002-0.04
times the speed of light, much faster than previous estimates derived using
only a reactive-diffusive description. Analytic solutions to hydrodynamical
jump conditions with a temperature dependent equation of state agree very well
with our numerical findings for fluid velocities. The most important effect of
neutrino cooling is that the conversion front stalls at lower density (below
approximately 2 times saturation density). In a 2-dimensional setting, such
rapid speeds and neutrino cooling may allow for a flame wrinkle instability to
develop, possibly leading to detonation.Comment: 5 pages, 3 figures (animations online at
http://www.capca.ucalgary.ca/~bniebergal/webPHP/research.php
Existence and approximation of probability measure solutions to models of collective behaviors
In this paper we consider first order differential models of collective
behaviors of groups of agents based on the mass conservation equation. Models
are formulated taking the spatial distribution of the agents as the main
unknown, expressed in terms of a probability measure evolving in time. We
develop an existence and approximation theory of the solutions to such models
and we show that some recently proposed models of crowd and swarm dynamics fit
our theoretic paradigm.Comment: 31 pages, 1 figur
Accurate discretization of advection-diffusion equations
We present an exact mathematical transformation which converts a wide class
of advection-diffusion equations into a form allowing simple and direct spatial
discretization in all dimensions, and thus the construction of accurate and
more efficient numerical algorithms. These discretized forms can also be viewed
as master equations which provides an alternative mesoscopic interpretation of
advection-diffusion processes in terms of diffusion with spatially varying
hopping rates
Quantum turbulence at finite temperature: the two-fluids cascade
To model isotropic homogeneous quantum turbulence in superfluid helium, we
have performed Direct Numerical Simulations (DNS) of two fluids (the normal
fluid and the superfluid) coupled by mutual friction. We have found evidence of
strong locking of superfluid and normal fluid along the turbulent cascade, from
the large scale structures where only one fluid is forced down to the vorticity
structures at small scales. We have determined the residual slip velocity
between the two fluids, and, for each fluid, the relative balance of inertial,
viscous and friction forces along the scales. Our calculations show that the
classical relation between energy injection and dissipation scale is not valid
in quantum turbulence, but we have been able to derive a temperature--dependent
superfluid analogous relation. Finally, we discuss our DNS results in terms of
the current understanding of quantum turbulence, including the value of the
effective kinematic viscosity
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