2,200 research outputs found
Relative-Periodic Elastic Collisions of Water Waves
We compute time-periodic and relative-periodic solutions of the free-surface
Euler equations that take the form of overtaking collisions of unidirectional
solitary waves of different amplitude on a periodic domain. As a starting
guess, we superpose two Stokes waves offset by half the spatial period. Using
an overdetermined shooting method, the background radiation generated by
collisions of the Stokes waves is tuned to be identical before and after each
collision. In some cases, the radiation is effectively eliminated in this
procedure, yielding smooth soliton-like solutions that interact elastically
forever. We find examples in which the larger wave subsumes the smaller wave
each time they collide, and others in which the trailing wave bumps into the
leading wave, transferring energy without fully merging. Similarities
notwithstanding, these solutions are found quantitatively to lie outside of the
Korteweg-de Vries regime. We conclude that quasi-periodic elastic collisions
are not unique to integrable model water wave equations when the domain is
periodic.Comment: 20 pages, 13 figure
Inf-sup estimates for the Stokes problem in a periodic channel
We derive estimates of the Babu\u{s}ka-Brezzi inf-sup constant for
two-dimensional incompressible flow in a periodic channel with one flat
boundary and the other given by a periodic, Lipschitz continuous function .
If is a constant function (so the domain is rectangular), we show that
periodicity in one direction but not the other leads to an interesting
connection between and the unitary operator mapping the Fourier sine
coefficients of a function to its Fourier cosine coefficients. We exploit this
connection to determine the dependence of on the aspect ratio of the
rectangle. We then show how to transfer this result to the case that is
or even by a change of variables. We avoid non-constructive
theorems of functional analysis in order to explicitly exhibit the dependence
of on features of the geometry such as the aspect ratio, the maximum
slope, and the minimum gap thickness (if passes near the substrate). We
give an example to show that our estimates are optimal in their dependence on
the minimum gap thickness in the case, and nearly optimal in the
Lipschitz case.Comment: 18 pages, 4 figure
An infinite branching hierarchy of time-periodic solutions of the Benjamin-Ono equation
We present a new representation of solutions of the Benjamin-Ono equation
that are periodic in space and time. Up to an additive constant and a Galilean
transformation, each of these solutions is a previously known, multi-periodic
solution; however, the new representation unifies the subset of such solutions
with a fixed spatial period and a continuously varying temporal period into a
single network of smooth manifolds connected together by an infinite hierarchy
of bifurcations. Our representation explicitly describes the evolution of the
Fourier modes of the solution as well as the particle trajectories in a
meromorphic representation of these solutions; therefore, we have also solved
the problem of finding periodic solutions of the ordinary differential equation
governing these particles, including a description of a bifurcation mechanism
for adding or removing particles without destroying periodicity. We illustrate
the types of bifurcation that occur with several examples, including degenerate
bifurcations not predicted by linearization about traveling waves.Comment: 27 pages, 6 figure
Program to determine space vehicle response to wind turbulence
Computer program was developed as prelaunch wind monitoring tool for Saturn 5 vehicle. Program accounts for characteristic wind changes including turbulence power spectral density, wind shear, peak wind velocity, altitude, and wind direction using stored variational statistics
A local construction of the Smith normal form of a matrix polynomial
We present an algorithm for computing a Smith form with multipliers of a
regular matrix polynomial over a field. This algorithm differs from previous
ones in that it computes a local Smith form for each irreducible factor in the
determinant separately and then combines them into a global Smith form, whereas
other algorithms apply a sequence of unimodular row and column operations to
the original matrix. The performance of the algorithm in exact arithmetic is
reported for several test cases.Comment: 26 pages, 6 figures; introduction expanded, 10 references added, two
additional tests performe
Museums & Society 2034: Trends and Potential Futures
What challenges will society and museums face in the next quarter-century? How will the demographic profile of America change between now and 2034? How will energy and infrastructure costs affect the sustainability of museums? What will Web 3.0 -- or 5.0 or 6.0 -- look like? Will the "real" survive the assault of the "virtual"? Will the number of leisure-time alternatives continue to grow? Will the lines between work and leisure, public and private, continue to blur? Most importantly, how will museums face these challenges and shape the future they will have to inhabit?This report, commissioned by the Center for the Future of Museums at the American Association of Museums, projects current social trends to 2034 and suggests how museums can face future challenges while continuing to meet their mission of public service. The report focuses on four major trends: demographic shifts, globalization, the revolution in information and communication technologies, and new cultural assumptions about the primacy of the individual as creator and curator
Variational Particle Schemes for the Porous Medium Equation and for the System of Isentropic Euler Equations
Both the porous medium equation and the system of isentropic Euler equations
can be considered as steepest descents on suitable manifolds of probability
measures in the framework of optimal transport theory. By discretizing these
variational characterizations instead of the partial differential equations
themselves, we obtain new schemes with remarkable stability properties. We show
that they capture successfully the nonlinear features of the flows, such as
shocks and rarefaction waves for the isentropic Euler equations. We also show
how to design higher order methods for these problems in the optimal transport
setting using backward differentiation formula (BDF) multi-step methods or
diagonally implicit Runge-Kutta methods.Comment: 36 pages, 9 figures; re-wrote introduction, added 6 references, added
discussion of diagonally implicit Runge-Kutta schemes, moved some material to
appendice
Comparison of five methods of computing the Dirichlet-Neumann operator for the water wave problem
We compare the effectiveness of solving Dirichlet-Neumann problems via the
Craig-Sulem (CS) expansion, the Ablowitz-Fokas-Musslimani (AFM) implicit
formulation, the dual AFM formulation (AFM*), a boundary integral collocation
method (BIM), and the transformed field expansion (TFE) method. The first three
methods involve highly ill-conditioned intermediate calculations that we show
can be overcome using multiple-precision arithmetic. The latter two methods
avoid catastrophic cancellation of digits in intermediate results, and are much
better suited to numerical computation.
For the Craig-Sulem expansion, we explore the cancellation of terms at each
order (up to 150th) for three types of wave profiles, namely band-limited,
real-analytic, or smooth. For the AFM and AFM* methods, we present an example
in which representing the Dirichlet or Neumann data as a series using the AFM
basis functions is impossible, causing the methods to fail. The example
involves band-limited wave profiles of arbitrarily small amplitude, with
analytic Dirichlet data. We then show how to regularize the AFM and AFM*
methods by over-sampling the basis functions and using the singular value
decomposition or QR-factorization to orthogonalize them. Two additional
examples are used to compare all five methods in the context of water waves,
namely a large-amplitude standing wave in deep water, and a pair of interacting
traveling waves in finite depth.Comment: 31 pages, 18 figures. (change from version 1: corrected error in
table on page 12
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