Research
My research revolves around numerical analysis, primarily in the study of computational fluid dynamics. My work to date has centered largely around the finite element method (FEM). These studies include the derivation of the energy (in) equality, stability of the model's numerical solution, error estimates, and simulations of benchmark fluid flow problems.
My current research focus is on the development of Data-Driven Galerkin (d2G) methods for turbulence. This involves Galerkin reduced-order models (G-ROMs) constructed from FEM full-order models (FOMs). At the core of this research is the idea of using spatial and spectral filtering methods for under-resolved flows, which can dramatically increase the ROM accuracy and stability.
Publications
Submitted
M. Neda, J. Reyes, and J. Waters, “Verman stabilization for nonlinear Greenshield’s model for traffic flow,” Addressing Modern Challenges in the Mathematical, Statistical, and Computational Sciences: AMMCS 2023 Proceeding, (Submitted).
S. Breckling, J. Fiordilino, J. Reyes, and S. Shields, “A note on the long-time stability of pressure solutions to the 2D Navier Stokes equation,” Applied Mathematics and Computations, (Minor Revisions)
Published
L. Davis, M. Neda, F. Pahlevani, J. Reyes, and J. Waters, “A numerical study of a stabilized hyperbolic equation inspired by models for bio-polymerization,” Computational Methods in Applied Mathematics, (2024), doi:10.1515/cmam-2023-0222
S. C. Huang, A. Johnson, M. Neda, J. Reyes, and H. Tehrani, “A generalization of the Smagorinsky model,” Applied Mathematics and Computation, vol. 469, p. 128 545, (2024), issn: 0096-3003. doi:10.1016/j.amc.2024.128545.
J.Reyes, " Examples of identities and inequalities for the nonlinear term in the Navier Stokes equation," Examples and Counterexamples, vol. 3, pp. 100-109, (2023), ISSN: 2666-657X. doi: /10.1016/j.exco.2023.100109
P. J.-S. Shiue, A. G. Shannon, S. C. Huang, and J. E. Reyes, “A generalized computation procedure for Ramanujan-type identities and cubic Shevelev sum,” Notes on Number Theory and Discrete Mathematics, vol. 29, no. 1, pp. 98–129, (2023). doi: 10.7546/nntdm.2023.29.1.98-129
S. Ingimarson, M. Neda, L. Rebholz, J. Reyes, and A. Vu, “Improved long time accuracy for projection methods for Navier-Stokes equations using EMAC formulation,”International Journal of Numerical Analysis & Modeling, vol. 20, no. 2, pp. 176–198, (2023).doi:10.4208/ijnam2023-1008
P. J.-S. Shiue, A. G. Shannon, S. C. Huang, and J. E. Reyes, “Notes on efficient computation of Ramanujan cubic equations,”Notes on Number Theory and Discrete Mathematics, vol. 28, no. 2, pp. 350–375, (2022).doi:10.7546/nntdm.2022.28.2.350-375
P. J. Shuie, S. C. Huang, and J. E. Reyes, “Algorithms for computing sums of powers of arithmetic progressions by using Eulerian numbers,” Notes on Number Theory and Discrete Mathematics, vol. 27, no. 4, pp. 140–148, (2021). doi: 10.7546/nntdm.2021.27.4.140-148.
Simulations
Flow Past Cylinder :
Reynold Number = 500
P2/P1 Taylor Hood Finite Elements
Crank-Nicolson Time Stepper
Example Mesh bellow (Mesh used in videos much more fine)
Vreman Stabilization :
P2 Finite Elements
χ (chi) - Tunable stabilization parameter.
Time Filtered Backward Euler Time Stepper
1D- LWR Traffic Model with Greenshields non-linearity