Research Projects

I am interested in developing mathematical models and computational algorithms for studying fluid behavior.
Post-Doctoral Projects

  • Scalable Methods for Convection-Diffusion Systems Numerical simulation of fluid dynamics allows for improved prediction and design of natural and engineered systems such as those involving water, oil, and blood. Such systems often involve dynamics that occur on disparate length and time scales due to variations in inertial and viscous forces. In order to numerically simulate these dynamics, complex mathematical modeling, and scalable computational methods are required. The principal goal of this project is to develop a scalable numerical framework to be used in conducting fluid flow simulations involving convection and diffusion. We are particularly interested in using these computational methods to investigate fluid behavior near critical points of multiphase flows in the presence of convection as part of the metrology mission at NIST.

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Graduate Projects in Scientific Computation

  • Fast Solvers for Models of Fluid Flow with Spectral Elements. We introduce a preconditioning technique based on Domain Decomposition and the Fast Diagonalization Method that can be applied to tensor product based discretizations of the steady convection-diffusion and the linearized Navier-Stokes equations. The method is based on iterative substructuring where fast diagonalization is used to efficiently eliminate the interior degrees of freedom and subsidiary subdomain solves. We demonstrate the effectiveness of this preconditioner in numerical simulations using a spectral element discretization.

    This work extends the use of Fast Diagonalization to steady convection-diffusion systems. We also extend the "least-squares commutator" preconditioner, originally developed for the finite element method, to a matrix-free spectral element framework. We show that these two advances, when used together, allow for efficient computation of steady-state solutions the the incompressible Navier-Stokes equations using high-order spectral element discretizations.

    This work has been presented at several seminars, and conferences. Here are some slides, posters and link to my disseration.

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  • Spectral Element Method for Burgers Equation with p-refinement. To numerically simulate the dynamic behavior of fluids it is important to have an efficient, accurate discretization of the governing equations. The spectral element method breaks the computational domain into multiple subdomains called elements, and defines a high order polynomial basis on each element to represent the flow variables. By changing the polynomial basis on each element based on flow characteristics, one can obtain an efficient, accurate discretization through refinement. For my advanced scientific computation course I implemented a refinement scheme for the spectral element method to discretize the one-dimensional Burgers equation and developed solvers for this discretization, as well as a two-dimensional spectral element discretization of the unsteady advection-diffusion equation.

    This work was presented to the Advanced Scientific Computation Course on May 14, 2004. During the summer of 2004, I worked with Dr. Tom Clune of NASA Goddard to implement a parallel version of the two dimensional code. This work was presented as a poster at the Inaugural Symposium of the Burgers Program for Fluid Dynamics, Nov. 18, 2004. I then extended this work, to include the incompressible Navier Stokes equations, which led to my Ph.D. candidacy examination topic which was Parallel Adaptive Spectral Element Scheme with Geophysical Flow Applications.

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    Undergraduate Projects in Differential Equations

    • Finding Periodic Solutions to Nonlinear Differential Equations via the Homotopy Method. An important problem in engineering is to determine a set of initial parameters of a system to obtain a desired result. When the system is nonlinear this can be a difficult problem, in many cases with no analytical solution. However, under certain assumptions of the non-linear system, these parameters can be determined numerically via a homotopy method. For our test system, we considered a nonlinear differential equation with periodic forcing, and determined the initial conditions that lead to periodic solutions. We followed the result of W. Li and Z. Shen for finding periodic solutions to Duffing's Equation, and extended their theorem to discover multiple periodic solutions in the phase plane.

      This work was submitted as an honors thesis, and then published in the International Journal of Mathematical Education in Science & Technology in 2001, with my advisor T.H. Fay. (This gave me an Erdos number of 3 (yay!)). I also submitted a student paper to the MAA Louisiana MS 78th Annual Section Meeting, and presented this work at the 10th Annual USA/USM Mini conference in Undergraduate Research. Additionally, I developed a graduate homework assignment in scientific computation based on this work which is posted below.

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