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

该项目旨在提高斯佩尔曼学院（Spelman College）本科生和数学系教师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。通过与数学计算与实验研究所（ICERM）的数值偏微分方程学期的合作，该项目将促进数值偏微分方程的研究合作，使斯佩尔曼学院接触到ICERM项目，并将为本科生提供数值偏微分方程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旋量分解的方程，旋量分解是将二元混合物分离成两相。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将探索结合机器学习技术来及时发展模型的潜力。总体而言，该项目将前沿研究与教育和多样性倡议相结合，将提高斯佩尔曼学院应用数学的研究能力，并鼓励黑人女性攻读应用数学研究生学位。该研究项目旨在开发一种高阶数值方法来解决Cahn-Hilliard方程，这是二元混合物中旋量分解的模型。目的是研究隐式-显式（IMEX）龙格-库塔方法中隐式和显式分量的最佳分离，从而得到准确稳定的解。PI首先将Shen的半隐式方法扩展为IMEX隐式中点规则。然后，PI将确定在三阶IMEX龙格-库塔对角线方案中相同的分裂是否有益。研究还探讨了使用混合模型方法的可变流动性情况下，结合恒定流动性模型。目标是选择最优的分割来产生可变的移动行为。此外，该项目将机器学习技术纳入混合模型时间演化。所有提出的方法的稳定性和准确性将被彻底研究。了解二元混合物中旋量分解的动力学在材料科学、化学和工程中具有重要的应用。发展高阶数值方法并研究其稳定性有助于模拟复杂物理现象的数值技术的进步。此外，将机器学习集成到数值框架中，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项目和hbcu卓越研究项目共同资助。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 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.