PRIMES: Enhancing Capacity for Research in Applied Mathematics at Spelman College

该项目旨在提高斯佩尔曼学院数学系本科生和教职员工在应用数学方面的研究能力。斯佩尔曼学院是一所历史悠久的黑人女子学院。通过与计算与实验数学研究所(ICERM)在数值偏微分方程式方面的学期合作，该项目将促进数值偏微分方程的研究合作，让斯佩尔曼的教师接触ICERM项目，并将提供本科生在数值偏微分方程方面的研究机会和课程开发。该项目的研究重点是开发更好的计算技术来求解模拟调幅分解的方程，调幅分解是将二元混合物分离成两个相。目标是使用混合模型方法创建高阶数值方法，并研究这些方法的稳定性。此外，PI将探索纳入机器学习技术的潜力，以及时发展模型。总体而言，该项目将尖端研究与教育和以多样性为重点的倡议相结合，将提高斯佩尔曼学院应用数学的研究能力，并将鼓励黑人女性攻读应用数学研究生学位。这一研究项目旨在开发一种求解Cahn-Hilliard方程的高阶数值方法，Cahn-Hilliard方程是一种二元混合物的调幅分解模型。目标是研究隐式-显式(IMEX)龙格-库塔方法中隐式分量和显式分量之间的最优分裂，以产生准确和稳定的解。PI First将沈的半隐式方法推广到IMEX隐式中点规则。然后，PI将确定相同的分裂在三阶对角IMEX Runge-Kutta格式中是否有益。这项研究还探索了使用混合模型的方法来处理可变流动性的情况，其中包括恒定流动性模型。目标是选择最优的分裂来产生可变的移动行为。此外，该项目还将机器学习技术融入到混合模型时间演化中。所有提出的方法的稳定性和准确性都将得到彻底的调查。了解二元混合物的调幅分解动力学在材料科学、化学和工程中有着重要的应用。发展高阶数值方法并研究其稳定性有助于发展模拟复杂物理现象的数值技术。此外，将机器学习整合到数值框架中，为提高模拟的准确性和效率开辟了途径。该项目由数学科学部基础设施计划和HBCU-卓越研究计划联合资助。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 project aims to enhance research capacity in applied mathematics for undergraduates and faculty in the Department of Mathematics at Spelman College, a historically Black college for women. Through engagement with the Institute for Computational and Experimental Research in Mathematics (ICERM) semester on Numerical Partial Differential Equations, the project will foster research collaborations in numerical PDEs, expose Spelman faculty to ICERM programs, and will offer undergraduate research opportunities and curriculum development in numerical PDEs. The project’s research focus is to develop better computational techniques to solve the equations that model spinodal decomposition, which is the separation of binary mixtures into two phases. The goal is to create higher order numerical approaches using mixed-model methods and to investigate the stability properties of these methods. Additionally, the PI will explore the potential of incorporating machine learning techniques to evolve the model in time. Overall, this project combines cutting-edge research with educational and diversity focused initiatives that will improve research capacity in applied math at Spelman College and will encourage Black women to pursue graduate degrees in applied mathematics. This research project seeks to develop a higher order numerical approach for solving the Cahn-Hilliard equation, a model for spinodal decomposition in binary mixtures. The goal is to investigate the optimal splitting between the implicit and explicit components in an implicit-explicit (IMEX) Runge-Kutta method that yields an accurate and stable solution. The PI firsts extends a semi-implicit approach by Shen to an IMEX implicit midpoint rule. Then the PI will determine if the same splitting is beneficial in a third order diagonally IMEX Runge-Kutta scheme. The research also explores using a mixed-model approach for the variable mobility case that incorporates the constant mobility model. The goal is to choose the optimal splittings to produce variable mobility behavior. Furthermore, the project is incorporating machine learning techniques into the mixed model time evolution. The stability and accuracy of all the proposed methods will be thoroughly investigated. Understanding the dynamics of spinodal decomposition in binary mixtures has significant applications in materials science, chemistry, and engineering. Developing higher order numerical methods and investigating their stability contributes to the advancement of numerical techniques for simulating complex physical phenomena. Moreover, the integration of machine learning into the numerical framework opens avenues for enhancing the accuracy and efficiency of the simulations.The project is funded jointly by the Infrastructure program of the Division of Mathematical Sciences and the HBCU-Excellence in Research Program.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.