2,232 research outputs found

    Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and Arnoldi-Chebyshev algorithm

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    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

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    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

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    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

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    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

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    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

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    The eigenvector centrality equation λx=A x\lambda x = A \, x 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 x=Ax÷x = A x^{\div}, where x÷x^{\div} is the vector whose entries are the reciprocal of those of xx. 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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