Presidential Young Investigator Award: Rapid Numerical Algorithms for Scientific Computation

点击翻译按钮获取中文摘要

One goal is to incorporate the Fast Multipole Method (FMM) into the molecular dynamics simulation package being developed by Charles Peskin and Tamar Schlick at the Courant Institute. In such particle simulations, the greatest computational burden has classically been that of evaluating the self-consistent electrostatic field induced by all of the partially charged individual atoms. This is precisely the problem addressed by the FMM. The hope is that certain inaccuracies in molecular dynamics calculations can be eliminated, yielding more physically reasonable results. The second major area of research is potential theory for the heat equation. Earlier work, in collaboration with John Strain, has yielded a scheme for the rapid evaluation of heat potentials. It is planned to devote substantial research time over the next few years to the examination of extensions of this scheme as well as its ramifications. The combination of potential theory and integral equations may provide a fundamental change in the way heat equations are solved, since spatially complex and time- dependent domains are handled without additional difficulty. It also appears likely that the method will extend in a natural way to variable coefficient and nonlinear parabolic partial differential equations. The third area of research is phase separation in material science. A new integral equation-based scheme has been constructed for the Laplace equation in multiply connected domains. It is planned to incorporate this method into a model of "Ostwald ripening," which follows the dynamic motion of the interface between two phases. This requires the solution of Laplace's equation at each step.