11,313 research outputs found
Continuous and discrete Clebsch variational principles
The Clebsch method provides a unifying approach for deriving variational
principles for continuous and discrete dynamical systems where elements of a
vector space are used to control dynamics on the cotangent bundle of a Lie
group \emph{via} a velocity map. This paper proves a reduction theorem which
states that the canonical variables on the Lie group can be eliminated, if and
only if the velocity map is a Lie algebra action, thereby producing the
Euler-Poincar\'e (EP) equation for the vector space variables. In this case,
the map from the canonical variables on the Lie group to the vector space is
the standard momentum map defined using the diamond operator. We apply the
Clebsch method in examples of the rotating rigid body and the incompressible
Euler equations. Along the way, we explain how singular solutions of the EP
equation for the diffeomorphism group (EPDiff) arise as momentum maps in the
Clebsch approach. In the case of finite dimensional Lie groups, the Clebsch
variational principle is discretised to produce a variational integrator for
the dynamical system. We obtain a discrete map from which the variables on the
cotangent bundle of a Lie group may be eliminated to produce a discrete EP
equation for elements of the vector space. We give an integrator for the
rotating rigid body as an example. We also briefly discuss how to discretise
infinite-dimensional Clebsch systems, so as to produce conservative numerical
methods for fluid dynamics
Variational Principles for Lagrangian Averaged Fluid Dynamics
The Lagrangian average (LA) of the ideal fluid equations preserves their
transport structure. This transport structure is responsible for the Kelvin
circulation theorem of the LA flow and, hence, for its convection of potential
vorticity and its conservation of helicity.
Lagrangian averaging also preserves the Euler-Poincar\'e (EP) variational
framework that implies the LA fluid equations. This is expressed in the
Lagrangian-averaged Euler-Poincar\'e (LAEP) theorem proven here and illustrated
for the Lagrangian average Euler (LAE) equations.Comment: 23 pages, 3 figure
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Digital Creativity Support for Original Journalism
The decline in circulations and revenues resulting from the digitalization of news production and consumption has led to a crisis in journalism.Journalists have less time to research, investigate and write original stories, leading to problems for our democratic processes and holding the powerful to account. This paper reports the architecture, features and rationale for new digital creativity support designed to support journalists to discover more original angles onstories. It also summarises the evaluation of the tool’s use in 3 newsrooms
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Solving linear programs without breaking abstractions
We show that the ellipsoid method for solving linear programs can be implemented in a way that respects the symmetry of the program being solved. That is to say, there is an algorithmic implementation of the method that does not distinguish, or make choices, between variables or constraints in the program unless they are distinguished by properties definable from the program. In particular, we demonstrate that the solvability of linear programs can be expressed in fixed-point logic with counting (FPC) as long as the program is given by a separation oracle that is itself definable in FPC. We use this to show that the size of a maximum matching in a graph is definable in FPC. This settles an open problem first posed by Blass, Gurevich and Shelah [Blass et al. 1999]. On the way to defining a suitable separation oracle for the maximum matching program, we provide FPC formulas defining canonical maximum flows and minimum cuts in undirected capacitated graphs.Research supported by EPSRC grant EP/H026835.This is the author accepted manuscript. The final version is available from ACM via http://dx.doi.org/10.1145/282289
Leray and LANS- modeling of turbulent mixing
Mathematical regularisation of the nonlinear terms in the Navier-Stokes
equations provides a systematic approach to deriving subgrid closures for
numerical simulations of turbulent flow. By construction, these subgrid
closures imply existence and uniqueness of strong solutions to the
corresponding modelled system of equations. We will consider the large eddy
interpretation of two such mathematical regularisation principles, i.e., Leray
and LANS regularisation. The Leray principle introduces a {\bfi
smoothed transport velocity} as part of the regularised convective
nonlinearity. The LANS principle extends the Leray formulation in a
natural way in which a {\bfi filtered Kelvin circulation theorem},
incorporating the smoothed transport velocity, is explicitly satisfied. These
regularisation principles give rise to implied subgrid closures which will be
applied in large eddy simulation of turbulent mixing. Comparison with filtered
direct numerical simulation data, and with predictions obtained from popular
dynamic eddy-viscosity modelling, shows that these mathematical regularisation
models are considerably more accurate, at a lower computational cost.Comment: 42 pages, 12 figure
An integrable shallow water equation with peaked solitons
We derive a new completely integrable dispersive shallow water equation that
is biHamiltonian and thus possesses an infinite number of conservation laws in
involution. The equation is obtained by using an asymptotic expansion directly
in the Hamiltonian for Euler's equations in the shallow water regime. The
soliton solution for this equation has a limiting form that has a discontinuity
in the first derivative at its peak.Comment: LaTeX file. Figure available from authors upon reques
Highly turbulent solutions of LANS-alpha and their LES potential
We compute solutions of the Lagrangian-Averaged Navier-Stokes alpha-model
(LANS) for significantly higher Reynolds numbers (up to Re 8300) than have
previously been accomplished. This allows sufficient separation of scales to
observe a Navier-Stokes (NS) inertial range followed by a 2nd LANS inertial
range. The analysis of the third-order structure function scaling supports the
predicted l^3 scaling; it corresponds to a k^(-1) scaling of the energy
spectrum. The energy spectrum itself shows a different scaling which goes as
k^1. This latter spectrum is consistent with the absence of stretching in the
sub-filter scales due to the Taylor frozen-in hypothesis employed as a closure
in the derivation of LANS. These two scalings are conjectured to coexist in
different spatial portions of the flow. The l^3 (E(k) k^(-1)) scaling is
subdominant to k^1 in the energy spectrum, but the l^3 scaling is responsible
for the direct energy cascade, as no cascade can result from motions with no
internal degrees of freedom. We verify the prediction for the size of the LANS
attractor resulting from this scaling. From this, we give a methodology either
for arriving at grid-independent solutions for LANS, or for obtaining a
formulation of a LES optimal in the context of the alpha models. The fully
converged grid-independent LANS may not be the best approximation to a direct
numerical simulation of the NS equations since the minimum error is a balance
between truncation errors and the approximation error due to using LANS instead
of the primitive equations. Furthermore, the small-scale behavior of LANS
contributes to a reduction of flux at constant energy, leading to a shallower
energy spectrum for large alpha. These small-scale features, do not preclude
LANS to reproduce correctly the intermittency properties of high Re flow.Comment: 37 pages, 17 figure
An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion
We study a class of 1+1 quadratically nonlinear water wave equations that
combines the linear dispersion of the Korteweg-deVries (KdV) equation with the
nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still
preserves integrability via the inverse scattering transform (IST) method.
This IST-integrable class of equations contains both the KdV equation and the
CH equation as limiting cases. It arises as the compatibility condition for a
second order isospectral eigenvalue problem and a first order equation for the
evolution of its eigenfunctions. This integrable equation is shown to be a
shallow water wave equation derived by asymptotic expansion at one order higher
approximation than KdV. We compare its traveling wave solutions to KdV
solitons.Comment: 4 pages, no figure
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