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

该项目旨在设计和研究具有奇异能量势的变温相场方程的高效、正性保持和熵稳定的数值格式。这项研究将提供对电池系统和其他能源设备中的两相流动和相分离的了解，在这些设备中，温度变化将显著影响系统的性能和耐用性。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STEM中代表性不足群体的参与，并加强新兴的跨学科研究生课程。结果将通过专著发布，算法和软件将免费向公众开放。更详细地说，该项目将开发数值格式，通过设计解决三个重要性质：正性保持、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--正性对于理论和数值分析来说都是一个重要而不平凡的问题。本研究将为一类综合的变温相场模型的建立提供理论和数值分析。该项目将是第一次尝试证明模式系统的任何数值格式的收敛。该数值方法将适用于大规模、多学科、多物理的科学模拟。这些算法和软件将影响多个领域的研究，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/转换的多相离子液体。预处理的概念对非线性科学问题至关重要，它可能会形成数据科学的新前沿。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.