Efficient, Adaptive, and Convergent Numerical Methods for Phase Field Equations with Applications

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This project focuses on the design and application of efficient computational algorithms for approximating the solutions to models describing various physical phenomena, including multi-phase fluid flow, tissue growth, phase transformations, and polymer processing. The investigator will study models that apply to applications that include optimizing organic photovoltaic devices for energy conversion; predicting cancerous tumor growth; simulating complex biological flows involving sub-cellular structures like endoplasmic reticulum; and designing durable semiconductor and energy storage materials. As a central pillar of this project, he will continue to develop his software package, BSAM, based on the efficient algorithms developed in the research. This software package, which is, and will always be, freely available and open source, is designed to solve a broad spectrum of nonlinear, multi-physics partial differential equations, in two and three dimensions. This tool is ultimately useful for researchers from many disciplines and provides the capability to efficiently simulate complex phenomena, without the need to reinvent algorithms or redesign code. The principal investigator will examine high-order, highly nonlinear partial differential equations through focus on three specific project goals. These include the design and rigorous numerical analysis of high-order energy stable numerical schemes, the design and analysis of algorithms and software for efficient two and three-dimensional time-space adaptive modeling and simulation, and the design and rigorous analysis of novel, nearly-optimally complex preconditioned nonlinear solvers.  The models under examination in the project describe a number of physical processes, including solidification; grain boundary dynamics; crack propagation; tumor growth; two-phase polymer flows; organic photovoltaic processing; and complex biological flows involving lipid bilayers. Because the equations under study are coupled systems of highly nonlinear, high-order partial differential equations, the analysis of their solutions and the design of efficient and reliable numerical methods that give rise to convergent approximations is a non-trivial task. The principal investigator will design unconditionally energy stable, second and third-order-in-time approximations and aims to rigorously prove that the schemes are optimally convergent. He will implement optimally or nearly-optimally efficient solvers that take advantage of the variational/convexity structure of the proposed schemes, producing sophisticated numerical software.