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Mamoudou Diane, Etzer Belfort
Technology is changing every day. The urge for
improvement or new version of any kind influences
the world to discover of new practice. For many
years, the Fast Fourier Transform (FFT) has
been the method in image manipulation. During this
summer, our research was to understand the
mathematical theory behind ``wavelets'' and determine
whether this new approach can replace FFT in the
contribution to the new release of Chalice,
a software package that does large image manipulation.
Peter E Fox, Erik Colnick
For the last several years students in the Applied Math courses,
Calculus 3 and Differential Equations, have been required to complete
computer labs. Up to this date, these labs have relied
upon the use of Matlab and Mathematica. Both of these software
packages require a considerable background knowledge to be able to use
them effectively. This has resulted in students spending more time
learning the software during the execution of the labs than actually
gaining useful insights into the mathematical concepts which the labs
are intended to demonstrate. It was our intention to create a
software package with a more intuitive interface that still provides
many of the tools that Matlab and Mathematica possess. The software
package we developed was written in Java to aid in portability of the
package to the various computer platforms which the students have
access to on campus.
This summer we have implemented both plotting and numerical tools
including: a 1D function plotter, a parametric plotter, 1D and 2D ODE
solvers, a contour plotter, a vector field plotter, a numerical integrator and
a root finder. We will explain some of the features of Java and demonstrate
some of the tools we have developed.
Jonathan Peeters
Given a surface profile of a water wave and the initial velocity
potential values along the surface, one can advance the wave in time
using conservation laws (Laplace' equation), pressure constraints
(Bernoulli's equation), and also the claim that particles on the surface
remain on the surface. Because the surface uniquely determines the
solution of Laplaces equation on the interior one can simply follow the
evolution of the surface in time. Given this advantage there has been
no shortage of methods developed to numerically analyze the propagation
of unsteady gravity waves, see for instance Dold [1]. For our purposes
we decided to use a conformal mapping scheme initially developed by
Fornberg [2]. This is an efficient technique that maps the deep-water
wave surface to a unit disk where Laplaces equation is easily solved by
FFT.
The problem with the method as is, is that due to its low
accuracy in time attempts to observe a water wave for a long period of
time (40 s) become unwieldy. To avoid this difficulty, I am presently
attempting to combine a technique that Dold [1] used with Fornberg's
code. The idea here is that the bulk of the numerical work is spent
attempting to find the mapping from the physical plane to the unit disk
where Laplaces equation can be solved easily. Once this mapping is
found repeated solutions of Laplaces equation for higher time
derivatives of the velocity potential and point velocities can be found
without great increases in computational effort. While this concept
seems obvious enough, the actual implementation and altering of the
present code is causing some difficulties. However, once the code is
set-up, we should be able to get as high as fourth-order accuracy in
time in a more efficient manner than the present second-order scheme
used by Fornberg.
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References
1. J.W. Dold, J. of Comput. Physics, 103, 90 (1992)
2. B. Fornberg, SIAM J. Sci. Stat. Comput. 1, 386 (1980)
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