Numerical Methods for Deterministic and Stochastic Phase Field Models

该项目将考虑移动界面问题的算法，这些问题在材料科学、几何、广泛的物理科学和工程中有广泛的应用。相场模型是描述运动界面问题的重要模型之一，它在多晶材料的成核和生长过程、化学反应、凝固动力学等许多跨学科应用中出现。本项目提出了几种准确高效的算法来求解相场模型，进一步探索不同区域之间的联系。此外，由于杂质的存在，随机性在相场模型的应用中起着重要的作用，本项目将对随机性相场模型的设计、分析和数值方法的实现进行研究。此外，这个项目训练研究生为他们未来的职业生涯提供必要的技能。本研究项目针对确定性相场模型和随机相场模型开发了一些精确而高效的数值算法。在确定性情况下，讨论了一般相场模型在不同空间中的最大误差边界及其对几何流的逼近，自适应双网格算法，以及四阶相场模型对拓扑变化具有鲁棒性的优化算法。为了更好地研究理论结果和预测其他新的结果，将开发一套相场模型和几何流。在随机情况下，重点讨论了四阶随机相场模型的离散稳定性和收敛性结果，以及利用随机抛物对偶性的二阶随机相场模型的离散稳定性和收敛性结果，为随机几何流的近似提供了理论基础。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 will consider algorithms for moving interface problems that have broad applications in materials science, geometry, and broad physical sciences and engineering. The phase field model is one of the most important models to formulate the moving interface problems, and it arises from many interdisciplinary applications, such as the nucleation and growth processes of polycrystalline materials, chemical reaction, solidification dynamics, and so on. This project proposes several accurate and efficient algorithms to solve the phase field models and to further explore the connections between different areas. In addition, the project will address randomness, which plays an important role in the phase field model applications due to the existence of the impurities, and specifically will consider the design, analysis, and implementation of numerical methods for the stochastic phase field models which incorporate the randomness. Moreover, this project trains graduate students to equip them with necessary skills for their future careers.This research project develops a few accurate and efficient numerical algorithms for both deterministic and stochastic phase field models. The deterministic case discusses the sharpest error bounds for the general phase field models in various spaces and their approximations to geometric flows, the adaptive two-grid algorithms, and some optimization algorithms which are robust to topological changes for the fourth-order phase field models. A package of the phase field models and geometric flows will be developed to better study the theoretical results and to predict other new results. The stochastic case focuses on discrete stability and convergence results for the fourth-order stochastic phase field models, and sharpest discrete stability and convergence results for the second-order stochastic phase field models using stochastic parabolic duality, which may provide theoretical foundation for the approximations to stochastic geometric flows.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.