909 research outputs found
Scalar Levin-Type Sequence Transformations
Sequence transformations are important tools for the convergence acceleration
of slowly convergent scalar sequences or series and for the summation of
divergent series. Transformations that depend not only on the sequence elements
or partial sums but also on an auxiliary sequence of so-called remainder
estimates are of Levin-type if they are linear in the , and
nonlinear in the . Known Levin-type sequence transformations are
reviewed and put into a common theoretical framework. It is discussed how such
transformations may be constructed by either a model sequence approach or by
iteration of simple transformations. As illustration, two new sequence
transformations are derived. Common properties and results on convergence
acceleration and stability are given. For important special cases, extensions
of the general results are presented. Also, guidelines for the application of
Levin-type sequence transformations are discussed, and a few numerical examples
are given.Comment: 59 pages, LaTeX, invited review for J. Comput. Applied Math.,
abstract shortene
Wilsonian renormalization, differential equations and Hopf algebras
In this paper, we present an algebraic formalism inspired by Butcher's
B-series in numerical analysis and the Connes-Kreimer approach to perturbative
renormalization. We first define power series of non linear operators and
propose several applications, among which the perturbative solution of a fixed
point equation using the non linear geometric series. Then, following
Polchinski, we show how perturbative renormalization works for a non linear
perturbation of a linear differential equation that governs the flow of
effective actions. Then, we define a general Hopf algebra of Feynman diagrams
adapted to iterations of background field effective action computations. As a
simple combinatorial illustration, we show how these techniques can be used to
recover the universality of the Tutte polynomial and its relation to the
-state Potts model. As a more sophisticated example, we use ordered diagrams
with decorations and external structures to solve the Polchinski's exact
renormalization group equation. Finally, we work out an analogous construction
for the Schwinger-Dyson equations, which yields a bijection between planar
diagrams and a certain class of decorated rooted trees.Comment: 42 pages, 26 figures in PDF format, extended version of a talk given
at the conference "Combinatorics and physics" held at Max Planck Institut
fuer Mathematik in Bonn in march 2007, some misprints correcte
Improvements to the construction of binary black hole initial data
Construction of binary black hole initial data is a prerequisite for
numerical evolutions of binary black holes. This paper reports improvements to
the binary black hole initial data solver in the Spectral Einstein Code, to
allow robust construction of initial data for mass-ratio above 10:1, and for
dimensionless black hole spins above 0.9, while improving efficiency for lower
mass-ratios and spins. We implement a more flexible domain decomposition,
adaptive mesh refinement and an updated method for choosing free parameters. We
also introduce a new method to control and eliminate residual linear momentum
in initial data for precessing systems, and demonstrate that it eliminates
gravitational mode mixing during the evolution. Finally, the new code is
applied to construct initial data for hyperbolic scattering and for binaries
with very small separation.Comment: 28 pages, 13 figures, 1 tabl
A new generalized domain decomposition strategy for the efficient parallel solution of the FDS-pressure equation. Part I: Theory, Concept and Implementation
Due to steadily increasing problem sizes and accuracy requirements as well as storage restrictions on single-processor systems, the efficient numerical simulation
of realistic fire scenarios can only be obtained on modern high-performance computers based on multi-processor architectures. The transition to those systems
requires the elaborate parallelization of the underlying numerical concepts which must guarantee the same result as a potentially corresponding serial execution and preserve the convergence order of the original serial method. Because
of its low degree of inherent parallelizm, especially the efficient parallelization of the elliptic pressure equation is still a big challenge in many simulation programs for fire-induced flows such as the Fire Dynamics Simulator (FDS). In order to avoid losses of accuracy or numerical instabilities, the parallelization process must definitely take into account the strong global character of the physical pressure. The current parallel FDS solver is based on a relatively coarse-grained parallellization concept which can’t guarantee these requirements in all cases.
Therefore, an alternative parallel pressure solver, ScaRC, is proposed which ensures a high degree of global coupling and a good computational performance at the same time. Part I explains the theory, concept and implementation of this
new strategy, whereas Part II describes a series of validation and verification tests to proof its correctness
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