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，几何学和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，它产生于多晶材料的成核和生长过程、化学反应、凝固动力学等多学科交叉的应用中，本项目提出了几种精确有效的算法来求解相场模型，并进一步探索不同领域之间的联系。此外，该项目将解决随机性，由于杂质的存在，它在相场模型应用中起着重要作用，特别是将考虑随机性相场模型的数值方法的设计，分析和实施。此外，本研究计画也培养研究生，使他们具备未来工作所需的技能。本研究计画针对确定性与随机性相场模型，发展出数种精确且有效率的数值演算法。确定性的情况下，讨论了尖锐的误差界的一般相场模型在各种空间和它们的近似几何流，自适应两个网格算法，和一些优化算法是强大的拓扑结构的变化，为四阶相场模型。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。随机情况下的重点是离散稳定性和收敛结果的四阶随机相场模型，和尖锐的离散稳定性和收敛结果的二阶随机相场模型使用随机抛物对偶，该奖项反映了NSF的法定使命，并通过使用基金会的学术价值和更广泛的影响评审标准。

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.