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

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。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。本项目将为其数值模拟研究一种有效的数值框架。所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理。该项目还将为K-12学生、本科生和研究生提供培训和外联机会，重点是数学建模、数值分析和工程应用，并特别注意促进在STEM中代表性不足的群体的参与。该项目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，将仔细构造和研究高阶辛Runge-Kutta方法，以确保能量稳定性。为了有效地捕捉陡峭的梯度并易于处理复杂的边界条件，空间离散技术将采用不连续Galerkin(DG)方法。将对经典模式DG方法和多分辨率DG方法的高维计算效率进行调查和比较。该奖项反映了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。

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.