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

本项目旨在设计和研究具有奇异能势的变温相场方程的高效、保正、熵稳定的数值格式。该研究将提供对电池系统和其他能源设备中两相流和相分离的理解，在这些设备中，温度变化将会显著影响系统的性能和耐久性。该项目将通过科学计算、建模和数值分析方面的前沿培训机会来加强研究生的培训。这项研究将提供机会，增加代表性不足群体对STEM的参与，并加强新兴的跨学科研究生项目。结果将通过专著传播，算法和软件将免费提供给公众。更详细地说，该项目将通过设计开发解决三个重要性质的数值方案：保正性、能量/熵稳定性和无条件唯一可解性。对于本文所研究的变温模型，即具有奇异势的梯度流动，正性是理论和数值分析的一个重要而非平凡的问题。本研究将为一类全面的变温相场模型提供理论数值分析和模型建立。该项目将首次尝试证明模型系统的任何数值格式的收敛性。数值方法将适用于大规模、多学科、多物理场的科学模拟。算法和软件将影响几个领域的研究，包括原子尺度的相变，复杂的生物生长和癌症，以及用于能量存储/转换的多相离子流体。预处理的概念对于非线性科学问题是至关重要的，它将潜在地形成数据科学的新前沿。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.