6,100 research outputs found
Solving Nonlinear Parabolic Equations by a Strongly Implicit Finite-Difference Scheme
We discuss the numerical solution of nonlinear parabolic partial differential
equations, exhibiting finite speed of propagation, via a strongly implicit
finite-difference scheme with formal truncation error . Our application of interest is the spreading of
viscous gravity currents in the study of which these type of differential
equations arise. Viscous gravity currents are low Reynolds number (viscous
forces dominate inertial forces) flow phenomena in which a dense, viscous fluid
displaces a lighter (usually immiscible) fluid. The fluids may be confined by
the sidewalls of a channel or propagate in an unconfined two-dimensional (or
axisymmetric three-dimensional) geometry. Under the lubrication approximation,
the mathematical description of the spreading of these fluids reduces to
solving the so-called thin-film equation for the current's shape . To
solve such nonlinear parabolic equations we propose a finite-difference scheme
based on the Crank--Nicolson idea. We implement the scheme for problems
involving a single spatial coordinate (i.e., two-dimensional, axisymmetric or
spherically-symmetric three-dimensional currents) on an equispaced but
staggered grid. We benchmark the scheme against analytical solutions and
highlight its strong numerical stability by specifically considering the
spreading of non-Newtonian power-law fluids in a variable-width confined
channel-like geometry (a "Hele-Shaw cell") subject to a given mass
conservation/balance constraint. We show that this constraint can be
implemented by re-expressing it as nonlinear flux boundary conditions on the
domain's endpoints. Then, we show numerically that the scheme achieves its full
second-order accuracy in space and time. We also highlight through numerical
simulations how the proposed scheme accurately respects the mass
conservation/balance constraint.Comment: 36 pages, 9 figures, Springer book class; v2 includes improvements
and corrections; to appear as a contribution in "Applied Wave Mathematics II
Deformation statistics of sub-Kolmogorov-scale ellipsoidal neutrally buoyant drops in isotropic turbulence
Small droplets in turbulent flows can undergo highly variable deformations
and orientational dynamics. For neutrally buoyant droplets smaller than the
Kolmogorov scale, the dominant effects from the surrounding turbulent flow
arise through Lagrangian time histories of the velocity gradient tensor. Here
we study the evolution of representative droplets using a model that includes
rotation and stretching effects from the surrounding fluid, and restoration
effects from surface tension including a constant droplet volume constraint,
while assuming that the droplets maintain an ellipsoidal shape. The model is
combined with Lagrangian time histories of the velocity gradient tensor
extracted from DNS of turbulence to obtain simulated droplet evolutions. These
are used to characterize the size, shape and orientation statistics of small
droplets in turbulence. A critical capillary number, is identified
associated with unbounded growth of one or two of the droplet's semi-axes.
Exploiting analogies with dynamics of polymers in turbulence, the number
can be predicted based on the large deviation theory for the largest Finite
Time Lyapunov exponent. Also, for sub-critical the theory enables
predictions of the slope of the power-law tails of droplet size distributions
in turbulence. For cases when the viscosities of droplet and outer fluid differ
in a way that enables vorticity to decorrelate the shape from the straining
directions, the large deviation formalism based on the stretching properties of
the velocity gradient tensor loses validity and its predictions fail. Even
considering the limitations of the assumed ellipsoidal droplet shape, the
results highlight the complex coupling between droplet deformation, orientation
and the local fluid velocity gradient tensor to be expected when small viscous
drops interact with turbulent flows
MORPH: A Reference Architecture for Configuration and Behaviour Self-Adaptation
An architectural approach to self-adaptive systems involves runtime change of
system configuration (i.e., the system's components, their bindings and
operational parameters) and behaviour update (i.e., component orchestration).
Thus, dynamic reconfiguration and discrete event control theory are at the
heart of architectural adaptation. Although controlling configuration and
behaviour at runtime has been discussed and applied to architectural
adaptation, architectures for self-adaptive systems often compound these two
aspects reducing the potential for adaptability. In this paper we propose a
reference architecture that allows for coordinated yet transparent and
independent adaptation of system configuration and behaviour
Explain3D: Explaining Disagreements in Disjoint Datasets
Data plays an important role in applications, analytic processes, and many
aspects of human activity. As data grows in size and complexity, we are met
with an imperative need for tools that promote understanding and explanations
over data-related operations. Data management research on explanations has
focused on the assumption that data resides in a single dataset, under one
common schema. But the reality of today's data is that it is frequently
un-integrated, coming from different sources with different schemas. When
different datasets provide different answers to semantically similar questions,
understanding the reasons for the discrepancies is challenging and cannot be
handled by the existing single-dataset solutions.
In this paper, we propose Explain3D, a framework for explaining the
disagreements across disjoint datasets (3D). Explain3D focuses on identifying
the reasons for the differences in the results of two semantically similar
queries operating on two datasets with potentially different schemas. Our
framework leverages the queries to perform a semantic mapping across the
relevant parts of their provenance; discrepancies in this mapping point to
causes of the queries' differences. Exploiting the queries gives Explain3D an
edge over traditional schema matching and record linkage techniques, which are
query-agnostic. Our work makes the following contributions: (1) We formalize
the problem of deriving optimal explanations for the differences of the results
of semantically similar queries over disjoint datasets. (2) We design a 3-stage
framework for solving the optimal explanation problem. (3) We develop a
smart-partitioning optimizer that improves the efficiency of the framework by
orders of magnitude. (4)~We experiment with real-world and synthetic data to
demonstrate that Explain3D can derive precise explanations efficiently
Four-Dimensional Yang-Mills Theory as a Deformation of Topological BF Theory
The classical action for pure Yang--Mills gauge theory can be formulated as a
deformation of the topological theory where, beside the two-form field
, one has to add one extra-field given by a one-form which transforms
as the difference of two connections. The ensuing action functional gives a
theory that is both classically and quantistically equivalent to the original
Yang--Mills theory. In order to prove such an equivalence, it is shown that the
dependency on the field can be gauged away completely. This gives rise
to a field theory that, for this reason, can be considered as semi-topological
or topological in some but not all the fields of the theory. The symmetry group
involved in this theory is an affine extension of the tangent gauge group
acting on the tangent bundle of the space of connections. A mathematical
analysis of this group action and of the relevant BRST complex is discussed in
details.Comment: 74 pages, LaTeX, minor corrections; to be published in Commun. Math.
Phy
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