Biography

I am a fourth-year Ph.D. candidate in the Department of Mathematics at the University of Alabama. My research focuses on numerical analysis, particularly discontinuous Galerkin finite element methods for time-dependent PDEs. I use Fourier and energy analysis to study the stability and accuracy of DG methods applied to convection-dominated, convection-diffusion, and dispersion problems. I am also interested in time integration methods such as strong stability preserving Runge–Kutta schemes. My current work involves improving CFL conditions through Fourier analysis of DG methods applied to the transport and heat equations.

Academic Status

PhD Student - 4th

Research Area/Department

Applied Mathematics

Major/Specialty

Applied Mathematics; Numerical Analysis with a focus on the discontinuous Galerkin Finite Element Method.

Degrees Earned or in Progress

MSc. Mathematical Sciences (Earned) M.A. Mathematics (Earned) Ph.D. Mathematics (in Progress)

Academic Preparation

Partial Differential Equations (PDEs) Intro. Scientific Computing Iterative Methods Linear Systems Boundary Value Problems Numerical PDEs Numerical Analysis Real Analysis Linear Algebra Numerical Linear Algebra Functional Analysis Theory: Differential Equations Intro Complex Variables

Research/Academic Interests

I am a fourth-year Ph.D. candidate in the Department of Mathematics at the University of Alabama. My research lies in the field of numerical analysis, with a focus on discontinuous Galerkin (DG) finite element methods for solving time-dependent partial differential equations (PDEs). Specifically, I am interested in the stability, accuracy, and efficiency of DG schemes when applied to convection-dominated, convection-diffusion, and convection-dispersion problems. A core component of my work involves the use of Fourier analysis and energy methods to rigorously assess the stability properties and convergence behavior of semi-discrete DG formulations. Given the semi-discrete nature of DG methods for time-dependent problems, I also explore time integration strategies that preserve desirable properties of the spatial discretization. In particular, I study strong stability preserving (SSP) Runge–Kutta methods, total variation diminishing (TVD) schemes, and general Runge–Kutta methods represented through Butcher tableaus. My current research focuses on improving the allowable Courant–Friedrichs–Lewy (CFL) number through a detailed Fourier analysis of DG methods applied to the one-dimensional transport and heat equations. By examining the spectral behavior of these schemes, I aim to design or recommend modifications that extend stability regions and enhance computational efficiency without sacrificing accuracy.

Computational and Data Science Areas

Applied Mathematics

Motivation

I hereby apply to be part of the Student Track of the Sustainable Research Program, where I hope to contribute to and grow through interdisciplinary research grounded in applied mathematics. As a Ph.D. candidate in Applied Mathematics at The University of Alabama, expected to graduate in May 2027, my research focuses on high-order numerical methods, particularly the discontinuous Galerkin (DG) method for solving hyperbolic partial differential equations (PDEs). These methods offer high-order accuracy, stability, and flexibility—making them ideal for addressing complex, real-world challenges. I am currently exploring DG applications to conservation laws, with emphasis on stability, order of accuracy, and error control. The things I enjoy most in my scientific life are math research with proofs and theorems, analyzing more techniques to solve PDEs, the practical application of PDEs to real-world problems, and contact with engineering. In the next five years, I aim to be a math researcher in an academic institution, national laboratory, or industry, helping to create or improve upon methodologies for solving PDEs efficiently for the advancement of science. The Student Research Program (SRP) presents an opportunity for me to expand my research work, apply it to real-world challenges, and collaborate in a research environment such as the Lawrence Livermore National Lab, the Oak Ridge National Lab, the Argonne National Lab, and many other academic institutions whose faculty members are affiliated with the SRP. I am interested in mentorship, working among a team of world-class researchers, and collaborating on projects with faculty or students whose research interests align with mine. My interest in SRP began after learning from colleagues in the 2023 cohort, whose impactful projects and conference presentations demonstrated the academic rigor and professional value of the program and the affiliated institutions and faculties they were connected to. One of my colleagues received the Best Paper Award twice in two different conference presentations. Although I could not apply at the time due to qualifying exams, I remained committed to joining a future cohort. I am especially eager to be mentored by researchers at national laboratories, universities, and industry research centers whose expertise aligns closely with my research interests. Academically, I have built a strong foundation through graduate coursework in Partial Differential Equations, Numerical Analysis, Scientific Computing, Numerical Linear Algebra, Real Analysis, and Functional Analysis. These courses have equipped me with the necessary tools and expertise to model, analyze, and implement numerical solutions to complex problems. I am conversant with the use of Python and MATLAB for computational modeling, numerical experiments, and data visualization. I am also looking forward to being introduced to more advanced programming languages, becoming proficient in computer programming, understanding the usage and meaning of certain PDEs, and facing more complex challenges in applied mathematics that will build me into a refined mathematician and researcher. Beyond technical growth, I am drawn to SRP’s emphasis on professional development and career readiness. The opportunity to present research findings at conferences will help refine my communication skills and expand my academic network. Working alongside experienced scientists, engineers, and interdisciplinary teams will provide real-world, hands-on experience and help me build a strong résumé for a future in research. Exposure to cutting-edge projects will enhance my ability to apply theory in practice and prepare me for challenges in both academic and industry settings. I am eager to contribute my background in numerical methods, strengthen my collaborative research skills, and gain experience in high-impact environments such as Lawrence Livermore, Oak Ridge, Los Alamos National Lab, or any other scientific industries and institutions. I am confident that acceptance into the SRP and being paired with a faculty member and a team of researchers will prepare me to become a more capable, adaptable, and career-ready researcher, equipped to contribute meaningfully to the scientific community. Thank you for your time and consideration.