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。相场法是一种用于研究不同流体，固体和气体混合物的界面动力学的成熟理论方法。本项目将研究一个有效的数值框架，其数值模拟。开发的算法将适用于一大类物理问题，从而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推广机会，重点是数学建模，数值分析和工程应用，并特别关注促进STEM中代表性不足的群体成员的参与。该项目致力于设计，分析和实现一个完全离散的数值框架相场模型在高维空间和任意计算几何。在标量辅助变量的框架下，高阶辛龙格-库塔方法将被仔细构造和研究，以确保能量稳定性。为了有效地捕捉急剧的梯度和容易地处理复杂的边界条件，间断伽辽金（DG）方法将被用作空间离散技术。调查和比较经典模式DG和多分辨率DG方法之间的高维计算效率将carried.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.