I'm involved in the SUU Math Department's PDE Research Group. Last year the team consisted of three faculty members and two students. We found a numerical solution to a convolution model for phase transitions and implemented the scheme using MatLab and Mathematica.
My research also involves modeling Mullins-Sekerka flow in three dimensions. Coupled differential equations are reformulated as a system of boundary integral equations, which is solved numerically using either an explicit or semi-implicit scheme. This has involved creating a computer program that, among other tasks, generates discretized surfaces and solves huge systems of linear equations. The result is the evolution of surfaces over time, which is shown graphically via Maple.Current areas of focus are effectively utilizing particle-cluster methods, reducing run time, and generating surface models with evenly spaced nodes. I’m also investigating the validity of the scheme with relaxed regularity assumptions, dealing with topology changes, analytically showing convergence of a mean curvature estimation method, and looking into whether the Mullins-Sekerka problem is stiff.
A Numerical Scheme for Mullins-Sekerka Flow in Three Space Dimensions Advisor Dr. Peter Bates
The Mullins-Sekerka problem involves modeling a binary material with two stable concentration phases. A coarsening process occurs, and large particles grow while the smaller particles eventually dissolve. Single particles become spherical. My joint research with Dr. Peter Bates describes this process in three-dimensions using boundary integrals, which then can be solved numerically. The evolution of the particles over time is simulated numerically.
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| A scientific person . . . will never believe that the results of his own attempts are final . |
| Albert Einstein , taken from The Quotable Einstein, collected and edited by Alice Calaprice |