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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.

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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.

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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.

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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