Current Projects (in reverse chronological order):

• Dynamical Systems Framework for Mixing and Segregation in Tumbled Granular Flows
  • Summary: Studied, mathematically, the case of a vanishingly thin flowing layer. Showed the limiting dynamics of the kinematic continuum model of the flow are piecewise isometries. Identified a new mixing mechanism ("streamline jumping") in the absence of the usual chaotic advection precursor (streamline crossing).
  • Collaborators: Profs. Julio Ottino and Richard Lueptow (Northwestern University), Prof. Stephen Wiggins (University of Bristol) and Dr. Rob Sturman (University of Leeds).
  • Funding: Walter P. Murphy Fellowship, Travel Grant (2008), Travel Grant (2009).
  • Presentations: 62^nd DFD Annual Meeting.
  • Publications: Phys. Rev. Lett., Chaos.
• Physical Dynamics of Solitary Waves
  • Summary: Derived coarse-grained laws governing the centers of coherent structures in nonlinear wave equations (specifically, the sine-Gordon equation). Showed that these coarse-grain dynamics are identical to those of classical elastic point particles ("quasi-particles"), except for the presence of "cross-mass" (cf. Mach's principle). Not only does this method generalize the classical wave-particle duality, it also allows for the quantization of nonlinear waves of non-integrable equations. Extended the approach to deformable quasi-particles.
  • Collaborators: Prof. Christo Christov (University of Louisiana at Lafayette).
  • Funding: Travel Grant (2007), Travel Award (2008).
  • Presentations: Dynamics Days 2007, 7^th AIMS Int'l Conference.
  • Publications: Phys. Lett. A, DCDS Supplement.
• High-Resolution Godunov-type Schemes for Conservation Laws on Unstructured Meshes
  • Summary: Designed a simple and genuinely two-dimensional nonoscillatory reconstruction, constructed a robust fully-discrete central scheme on unstructured triangular grids using the latter, and benchmarked the scheme in 2D against a wide range of scalar and systems of hyperbolic conservation laws. Applied the genuinely multidimensional reconstruction, within the framework of higher-order upwind schemes on unstructured Voronoi meshes, to the simulation of multiphase flow in oil reservoirs.
  • Collaborators: Prof. Bojan Popov (Texas A&M University) and Dr. Ilya D. Mishev (ExxonMobil Upstream Research Company).
  • Funding: Regents Fellowship, Research Assistantship (through NSF Grant DMS-0510650), ExxonMobil Internship.
  • Presentations: 2006 SIAM Conference on Analysis of PDEs, 8^th IMACS Int'l Symposium on Iterative Methods, ULL Mathematics Colloquium, ExxonMobil.
  • Publications: ISC Technical Report, J. Comput. Phys., IAMCS Technical Report, HYP2008 Proceedings.
• Shock Formation in Homentropic (and Isentropic) Lossless Media
  • Summary: Developed a new numerical approach to the solution of the weakly-nonlinear theories of shock formation in perfect gases and, more generally, a class of barotropic fluids. Derived the consistent small-Mach-number finite-amplitude approximation of the Euler equations. Established a number of exact results on acceleration waves using singular surface theory. Benchmarked the model equations (i.e., the Lighthill–Westervelt, the inviscid Kuznetsov and the new weakly-nonlinear equation) numerically in a 1D shock-formation context. See also Waves in a Shock Tube.
  • Collaborators: Dr. Pedro Jordan (Naval Research Laboratory) and Prof. Christo Christov (University of Louisiana at Lafayette).
  • Funding: NRL Student Summer Employment Program, ASEE/ONR Naval Research Enterprise Intern Program.
  • Presentations: 6^th Int'l Conference NM&A.
  • Publications: Phys. Lett. A, CMES, QJMAM.
• Nonlinear Fourier Analysis of Internal Solitary Waves in the Ocean
  • Summary: Developed a version of the numerical periodic, inverse scattering transform for the KdV equation (Osborne's "nonlinear Fourier analysis") for the study of internal solitary waves generation in the ocean. Applied this technique to the anomalous signal loss problem in underwater acoustics. Undertook the first stp in understanding the propagation and evolution (e.g., the "birth," "aging" and "extinction") of internal solitary waves generated by tidal forcing over ocean bottom topography.
  • Collaborators: Drs. Stanley Chin-Bing and Alex Warn-Varnas (Naval Research Laboratory) and Dr. Jim Hawkins (Planning Systems Inc.).
  • Funding: NRL Student Summer Employment Program, ASEE/ONR Naval Research Enterprise Intern Program, Travel Award.
  • Presentations: NRL, NRL-SSC, 5^th IMACS Int'l Conference on Nonlinear Evolution Equations (Talk 1, Talk 2), EGU General Assembly 2007.
  • Publications: MTS/IEEE Proceedings, Lect. Notes in Earth Sci., Math. Comput. Simulat..

Selected Past Projects (in reverse chronological order):

• Solving Nonlinear Wave Equations using deal.II — This was a research project for the class Computational Software for Large-Scale PDE Solvers (MATH 664+694, Spring 2006, at Texas A&M). Using the (classical) finite element method (FEM) and the θ-method time discretization, I wrote a program, based on the deal.II FEM library, that solves the Neumann problem for the sine-Gordon soliton equation in 1, 2 and 3D. My project resulted in a tutorial program that is now part of the library.
• Breaking of Internal Waves in Lakes (joint work with Prof. Roman Stocker of MIT) — The goal of this project was to develop theoretical and numerical tools for the study and simulation of the nonlinear evolution of internal waves in stratified, rotating fluids (e.g., lakes on the earth's surface). The governing equations were formulated, and a Godunov-type (shock-capturing, finite-volume) scheme was developed to solve the equations numerically in 1D and 2D. This work was funded by MIT’s Undergraduate Research Opportunities Program (UROP).
• Wavelet-Galerkin Methods for Nonlinear Partial Differential Equations — This was a research project I worked on during the summer of 2004 at the Texas A&M Wavelets REU. The goal was to apply the wavelet-Galerkin method (see below) to the Korteweg–de Vries (nonlinear evolution) equation. A final report of my research is available in PDF.
• Waves in a Shock Tube — This was a research project for the class Nonlinear Dynamics II: Continuum Mechanics (18.354J, Spring 2004) at MIT. I derived the equations of gas dynamics from the Navier–Stokes equations for one dimensional unsteady flow. Under the isentropic gas assumption the pair of nonlinear coupled PDEs can be decoupled via diagonalization of the system matrix to obtain the "Riemann invariants." Then, I solved the piston problem for a receding, advancing and oscillating piston. A much-improved version of the final report of the research project has since been published in the MIT Undergraduate Journal of Mathematics.
• Wavelet Methods for Image Anti-Aliasing — This was an attempt at anti-aliasing images by wavelet-edge dectection and frequency filtering. Preliminary results can be found in the last section of the report entitled "Multiscale Image Edge Detection," which can be found below.
• Multiscale Image Edge Detection — This was a research project for the class Wavelets, Filter Banks and Applications (1.130/18.327, Spring 2004) at MIT. It discusses the multiscale generalization of the Canny edge detector using wavelets. I propose a method for thresholding the modulus maxima using the correlation coefficients of the modulus maxima across different resolutions for better edge recognition. I also discuss an application of the multiscale edge detector to the aliasing problem in computer generated images. A final report of my research is available in PDF.
• Crossing Number of a Graph — This was a research project for the class Principles of (Discrete) Applied Mathematics (18.310, Fall 2003) at MIT. It is an overview of the well-known Crossing Number problem and various results regarding it. I prove a few elementary results, but overall the paper is just a summary of the results in the literature on this topic. A final report of my research is available in PDF.
• Wavelet-Galerkin Methods for Partial Differential Equations — This was a research project I worked on during the summer of 2003 at the Texas A&M Wavelets REU. Although I do not discuss any novel methods, the report provides an overview of the subject. A final report of my research is available in PDF.

Disclaimer: Please do not copy my work and pass it off as your own. All research papers above have references to the original papers ands books from which the ideas come; likewise, if there are original ideas in the reports above, publication information, which can be cited, is provided. If all else fails, credit can be given by citing the research reports above as internet sources, if ideas from them must be used. Thanks!