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

Spelman College, Atlanta GA

Investigators

Abstract

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.