LEAPS-MPS: Robust and High Order Numerical Simulation for Phase Field Modeling

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。相场法是一种成熟的理论方法，用于研究不同流体、固体和气体混合物的界面动力学。本项目将研究一种有效的数值模拟框架。所开发的算法将适用于大量的物理问题，因此将有助于理解材料科学、生物学和其他相关领域的基本机制。该项目还将为K-12学生、本科生和研究生提供培训和拓展机会，重点是数学建模、数值分析和工程应用，并特别关注促进STEM中代表性不足群体成员的参与。本课题致力于为高维空间和任意计算几何中的相场模型设计、分析和实现一个完全离散的数值框架。在标量辅助变量框架下，认真构造和研究高阶辛龙格-库塔方法以保证能量稳定性。为了有效地捕捉尖锐梯度和方便地处理复杂的边界条件，将采用不连续伽辽金（DG）方法作为空间离散化技术。对经典模态DG和多分辨率DG方法的高维计算效率进行了研究和比较。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。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).The phase-field method is an established theoretical approach used to study the interfacial dynamics of the mixture of different fluids, solids, and gas. This project will investigate an efficient numerical framework for its numerical simulation. The algorithms developed will be applicable to a large class of physical problems, and thus will contribute to the understanding of fundamental mechanisms in materials science, biology, and other related fields. The project will also provide training and outreach opportunities to K-12 students, undergraduates, and graduate students, with a focus in mathematical modeling, numerical analysis, and engineering applications, and special attention to promoting the participation of members of groups underrepresented in STEM. This project is devoted to designing, analyzing, and implementing a fully discrete numerical framework for phase-field models in high dimensional spaces and arbitrary computational geometries. Under the scalar auxiliary variable framework, high order symplectic Runge-Kutta methods will be carefully constructed and studied to ensure energy stability. To efficiently capture the sharp gradient and easily handle the complex boundary conditions, the discontinuous Galerkin (DG) method will be employed as the spatial discretization technique. Investigation and comparison of high dimensional computational efficiency between classical modal DG and multiresolution DG methods will be carried out.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.