This proposal involves three projects where each one builds from prior NSF support. This first project involves the simulation of grain boundary motion in two and three dimensions using our recently developed multiphase variational level set framework. It is planned to extend this formulation to allow for the possibility of arbitrary surface tension between grains. This will be accomplished by combining this formulation with minimizing movements. The second project concerns the efficient computation the semi-classical limit of the Schroedinger equation. The proposed algorithm is based on the observation that if one transforms the Schroedinger equation using a transformation inspired by Gaussian wave packets, one arrives at another Schroedinger equation that is much more amenable to computation in the semiclassical limit project. The third project concerns modeling and efficient simulation of epitaxial growth using kinetic Monte Carlo (KMC). We plan to model and simulate both liquid drop epitaxy and heteroepitaxial growth. In each of these cases we aim to develop highly efficient KMC algorithms to allow simulation on time and lengths not previously possible, thereby facilitating model development and allowing comparison with experiments.
Each of the proposed projects has the potential to have a significant impact on problems that are both fundamental and technologically important. Epitaxial growth is scientifically interesting since it has effects on both nanoscales and mesoscales. It is technologically relevant since quantum dot materials are made in this way. Our proposed techniques will greatly increase the simulation speed thereby facilitating model development. The study of grain boundary motion using curvature flow is a classic problem in applied and computational mathematics which has importance in material science since various properties of polycrystalline substances can crucially depend on the details of the grain patterns. The fast simulation of the semi-classical limit of the Schroedinger equation could provide deeper insight into chemical reaction dynamics, molecular-surface scattering, and photodissociation, for example.