1,651 research outputs found
Coarsening Strategies for Unstructured Multigrid Techniques with Application to Anisotropic Problems
Over the years, multigrid has been demonstrated as an efficient technique for solving inviscid flow problems. However, for viscous flows, convergence rates often degrade. This is generally due to the required use of stretched meshes (i.e., the aspect ratio AR = Δy/Δx < < 1) in order to capture the boundary layer near the body. Usual techniques for generating a sequence of grids that produce proper convergence rates on isotropic meshes are not adequate for stretched meshes. This work focuses on the solution of Laplace's equation, discretized through a Galerkin finite-element formulation on unstructured stretched triangular meshes. A coarsening strategy is proposed and results are discussed
The Kernel Polynomial Method
Efficient and stable algorithms for the calculation of spectral quantities
and correlation functions are some of the key tools in computational condensed
matter physics. In this article we review basic properties and recent
developments of Chebyshev expansion based algorithms and the Kernel Polynomial
Method. Characterized by a resource consumption that scales linearly with the
problem dimension these methods enjoyed growing popularity over the last decade
and found broad application not only in physics. Representative examples from
the fields of disordered systems, strongly correlated electrons,
electron-phonon interaction, and quantum spin systems we discuss in detail. In
addition, we illustrate how the Kernel Polynomial Method is successfully
embedded into other numerical techniques, such as Cluster Perturbation Theory
or Monte Carlo simulation.Comment: 32 pages, 17 figs; revised versio
Extended Gravity Cosmography
Cosmography can be considered as a sort of a model-independent approach to
tackle the dark energy/modified gravity problem. In this review, the success
and the shortcomings of the CDM model, based on General Relativity and
standard model of particles, are discussed in view of the most recent
observational constraints. The motivations for considering extensions and
modifications of General Relativity are taken into account, with particular
attention to and theories of gravity where dynamics is
represented by curvature or torsion field respectively. The features of
models are explored in metric and Palatini formalisms. We discuss the
connection between gravity and scalar-tensor theories highlighting the
role of conformal transformations in the Einstein and Jordan frames.
Cosmological dynamics of models is investigated through the
corresponding viability criteria. Afterwards, the equivalent formulation of
General Relativity (Teleparallel Equivalent General Relativity) in terms of
torsion and its extension to gravity is considered. Finally, the
cosmographic method is adopted to break the degeneracy among dark energy
models. A novel approach, built upon rational Pad\'e and Chebyshev polynomials,
is proposed to overcome limits of standard cosmography based on Taylor
expansion. The approach provides accurate model-independent approximations of
the Hubble flow. Numerical analyses, based on Monte Carlo Markov Chain
integration of cosmic data, are presented to bound coefficients of the
cosmographic series. These techniques are thus applied to reconstruct
and functions and to frame the late-time expansion history of the
universe with no \emph{a priori} assumptions on its equation of state. A
comparison between the CDM cosmological model with and
models is reported.Comment: 82 pages, 35 figures. Accepted for publication in IJMP
QMR: A Quasi-Minimal Residual method for non-Hermitian linear systems
The biconjugate gradient (BCG) method is the natural generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. A novel BCG like approach is presented called the quasi-minimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a look-ahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported
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