2,232 research outputs found
Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and Arnoldi-Chebyshev algorithm
An efficient way of solving 2D stability problems in fluid mechanics is to
use, after discretization of the equations that cast the problem in the form of
a generalized eigenvalue problem, the incomplete Arnoldi-Chebyshev method. This
method preserves the banded structure sparsity of matrices of the algebraic
eigenvalue problem and thus decreases memory use and CPU-time consumption.
The errors that affect computed eigenvalues and eigenvectors are due to the
truncation in the discretization and to finite precision in the computation of
the discretized problem. In this paper we analyze those two errors and the
interplay between them. We use as a test case the two-dimensional eigenvalue
problem yielded by the computation of inertial modes in a spherical shell. This
problem contains many difficulties that make it a very good test case. It turns
out that that single modes (especially most-damped modes i.e. with high spatial
frequency) can be very sensitive to round-off errors, even when apparently good
spectral convergence is achieved. The influence of round-off errors is analyzed
using the spectral portrait technique and by comparison of double precision and
extended precision computations. Through the analysis we give practical recipes
to control the truncation and round-off errors on eigenvalues and eigenvectors.Comment: 15 pages, 9 figure
High Performance Computing for Stability Problems - Applications to Hydrodynamic Stability and Neutron Transport Criticality
In this work we examine two kinds of applications in terms of stability and perform numerical evaluations and benchmarks on parallel platforms. We consider the applicability
of pseudospectra in the field of hydrodynamic stability to obtain more information than a
traditional linear stability analysis can provide. Furthermore, we treat the neutron transport criticality problem and highlight the Davidson method as an attractive alternative to the so far widely used power method in that context
Metastability of solitary roll wave solutions of the St. Venant equations with viscosity
We study by a combination of numerical and analytical Evans function
techniques the stability of solitary wave solutions of the St. Venant equations
for viscous shallow-water flow down an incline, and related models. Our main
result is to exhibit examples of metastable solitary waves for the St. Venant
equations, with stable point spectrum indicating coherence of the wave profile
but unstable essential spectrum indicating oscillatory convective instabilities
shed in its wake. We propose a mechanism based on ``dynamic spectrum'' of the
wave profile, by which a wave train of solitary pulses can stabilize each other
by de-amplification of convective instabilities as they pass through successive
waves. We present numerical time evolution studies supporting these
conclusions, which bear also on the possibility of stable periodic solutions
close to the homoclinic. For the closely related viscous Jin-Xin model, by
contrast, for which the essential spectrum is stable, we show using the
stability index of Gardner--Zumbrun that solitary wave pulses are always
exponentially unstable, possessing point spectra with positive real part.Comment: 42 pages, 9 figure
Existence and stability of viscoelastic shock profiles
We investigate existence and stability of viscoelastic shock profiles for a
class of planar models including the incompressible shear case studied by
Antman and Malek-Madani. We establish that the resulting equations fall into
the class of symmetrizable hyperbolic--parabolic systems, hence spectral
stability implies linearized and nonlinear stability with sharp rates of decay.
The new contributions are treatment of the compressible case, formulation of a
rigorous nonlinear stability theory, including verification of stability of
small-amplitude Lax shocks, and the systematic incorporation in our
investigations of numerical Evans function computations determining stability
of large-amplitude and or nonclassical type shock profiles.Comment: 43 pages, 12 figure
Predictions for Scientific Computing Fifty Years from Now
This essay is adapted from a talk given June 17, 1998 at the conference "Numerical Analysis and Computers - 50 Years of Progress" held at the University of Manchester, England in commemoration of the 50th anniversary of the Mark 1 computer
A theory on power in networks
The eigenvector centrality equation is a successful
compromise between simplicity and expressivity. It claims that central actors
are those connected with central others. For at least 70 years, this equation
has been explored in disparate contexts, including econometrics, sociometry,
bibliometrics, Web information retrieval, and network science. We propose an
equally elegant counterpart: the power equation , where
is the vector whose entries are the reciprocal of those of . It
asserts that power is in the hands of those connected with powerless others. It
is meaningful, for instance, in bargaining situations, where it is advantageous
to be connected to those who have few options. We tell the parallel, mostly
unexplored story of this intriguing equation
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