Collaborative Research: Accurate and Structure-Preserving Numerical Schemes for Variable Temperature Phase Field Models and Efficient Solvers

The project aims to design and study highly efficient, positivity-preserving, and entropy-stable numerical schemes for variable-temperature phase field equations with singular energy potentials. The research will provide an understanding of two-phase flows and phase separation in battery systems and other energy devices where temperature variation would be significant, affecting the performance and durability of the systems. The project will enhance graduate student training through cutting-edge training opportunities in scientific computing, modeling, and numerical analysis. The research will provide opportunities to increase participation from underrepresented groups in STEM and enhance the emerging interdisciplinary graduate program. The results will be disseminated through a monograph, and the algorithms and software will be freely available to the public. In more detail, the project will develop numerical schemes addressing three important properties by design: positivity-preserving, energy/entropy stability, and unconditionally unique solvability. For the variable-temperature models studied herein – namely, gradient flows with singular potentials – positivity is an important and nontrivial issue for both theoretical and numerical analyses. This research will provide theoretical numerical analysis alongside of model building for a comprehensive class of variable-temperature phase field models. The project will be the first attempt to prove the convergence of any numerical scheme for the model systems. The numerical methods will be applicable in large-scale, multi-discipline, multi-physics scientific simulations. The algorithms and software will impact research in several areas, including atomic-scale phase transitions, complex biological growth and cancer, and multi-phase ionic fluids used in energy storage/conversion. The concept of preconditioning, which is vital for nonlinear scientific problems, will potentially form a new frontier in data science. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.