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

该项目旨在提高Spelman学院数学系本科生和教师的应用数学研究能力，Spelman学院是一所历史上的黑人女子学院。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，使斯佩尔曼教师接触ICERM课程，并将提供本科生研究机会和数值偏微分方程课程开发。该项目的研究重点是开发更好的计算技术，以解决模拟旋节分解的方程，即将二元混合物分离成两相。我们的目标是使用混合模型方法创建更高阶的数值方法，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的潜力，以及时发展模型。 总的来说，该项目结合了前沿研究与教育和多样性为重点的举措，将提高研究能力，在应用数学在斯佩尔曼学院，并将鼓励黑人妇女追求应用数学的研究生学位。本研究计划旨在发展高阶数值方法来求解Cahn-Hilliard方程，这是一个二元混合物中的旋节分解模型。 我们的目标是调查隐式和显式组件之间的隐式显式（IMEX）龙格库塔方法，产生一个准确和稳定的解决方案的最佳分裂。PI首先将Shen的半隐式方法推广到IMEX隐式中点规则。 然后PI将确定在三阶对角IMEX Runge-Kutta格式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的情况下，结合恒定的流动性模型。 我们的目标是选择最佳的分裂产生可变的流动性行为。 此外，该项目正在将机器学习技术纳入混合模型时间演化。 所有建议的方法的稳定性和准确性将进行彻底调查。 了解二元混合物中的旋节线分解动力学在材料科学、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，委员会认为，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和HBCU的基础设施计划共同资助，卓越研究计划。该奖项反映了NSF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进行评估，被认为值得支持的搜索.

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.