Numerical Methods for Multiphase Systems with Applications to Biology

This dissertation is comprised of a variety of efforts towards the development of fast numerical methods and their applications, particularly in the context of simulating biological systems. Scientific computing of these problems requires many considerations bridging gaps between computer science, applied mathematics, and the biology of the specific application. This dissertation spans these fields, with the formulation of heterogeneous mixture descriptions in one chapter, the study and implementation of efficient and robust numerical techniques in the next, and the application of this modeling framework and computational procedure to specific biological problems in the remaining chapters. The first of these efforts is the construction of multiphase models for macroscopic descriptions of biophysical problems. The second is the development of fast and flexible methods for simulating models derived from this modeling framework. The third is the revelation of this modeling framework to exhibit spatio-temporal patterns that can be initiated by localized perturbations in space. The fourth is the simulation of a four-phase model of biofilm formation implicated in Pierce's Disease.