Currently I am investigating numerical schemes for the nonlocal Allen-Cahn equation, Allen-Cahn equation, and other convolution models for phase transitions which have had much theoretical work done but little numerical work. These equations may model a variety of physical and biological phenomena involving media with properties varying in space, such as spinodal decomposition, Ostwald ripening, grain boundary motion, and the motion of antiphase boundaries in crystalline solids. The finite difference schemes are implemented in two and three dimensions using MatLab. Proving convergence and stability of the schemes is also part of the research. This is joint work with Jianlong Han of Southern Utah University.
A Numerical Scheme for Mullins-Sekerka Flow in Three Space Dimensions Advisor Dr. Peter Bates
Previous research involved modeling Mullins-Sekerka flow in three dimensions. The Mullins-Sekerka problem arises in modeling a binary material with two stable concentration phases. A coarsening process occurs, and large particles grow while smaller particles eventually dissolve. In my research, 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 involved creating a computer program that generates discretized surfaces and solves huge systems of linear equations. The result is the evolution of surfaces over time, shown graphically via Maple. Since the problem doesn't have an explicit solution even though theoretically a solution has been proven to exist, finding the solutions numerically allows us to see actual behavior.