Math 621 - Numerical Methods for Partial Differential Equations
Fall 2010 - Matthias K. Gobbert
Presentations of the Class Projects

  1. Friday, December 17, 2010, 01:05-01:15
    Modeling of Intracellular Calcium Waves and Sparks in one Spatial Dimension
    Zana Coulibaly, Department of Mathematics and Statistics
    A model for the flow of calcium on the scale of one heart cell is given by a system of time-dependent reaction-diffusion equations coupled by non-linear stochastic reaction terms and discrete release site. In this project, we turn the model from stochastic to deterministic and use finite difference to simulate the behavior of the system on different mesh size. The results we obtain suggests that solving the problem on finer meshes results in solving a different problem than having a smoother approximation to the solution. This work is joint with advisor Dr. Brad Peercy.

  2. Friday, December 17, 2010, 01:20-01:30
    Numerical Solution of a Distributed Optimal Control Problem with Semilinear Elliptic PDE Constraints Using Adjoint Methods
    Jyoti Saraswat, Department of Mathematics and Statistics
    In this project we solve distributed optimal control problem with semilinear elliptic PDE constraints. Our aim is to find the optimal control variable and the state variable which minimizes the cost functional. Each cost functional evaluation requires a nonlinear PDE solve. In order to solve the optimal control problem we use Newton's method which requires the gradient and the Hessian. The non trivial approach here is the use of adjoint methods to obtain the expressions for the gradient and the Hessian. This work is joint with advisor Dr. Andrei Draganescu.

  3. Friday, December 17, 2010, 01:35-01:45
    Performance Characteristics of Linear Solvers Using ODE15S in MATLAB
    Eric Katzaman, Department of Physics
    It is commonly known that MATLAB contains linear solver algorithms capable of solving partial differential equations using the Method of Lines approach; however, best practices associated with using these linear solvers are not as well known. We present performance characteristics on a solved system of partial differential equations with known analytical solution using the Jacobian option in conjunction with the NDFk and BDFk methods in ode15s. A convergence study using the lowest acceptable tolerance threshold is also reported. This work is joint with instructor Dr. Matthias Gobbert.

  4. Friday, December 17, 2010, 01:50-02:00
    Numerical Solution of an Elliptic PDE using Finite Differences on a Staggered Grid
    David Stanley, Department of Mathematics and Statistics
    In this paper we will solve a 2-D elliptic PDE using the Finite Differences method. However, this PDE will contain a spatially dependent diffusivity coefficient inside the divergence term which we will take into account by using a staggered grid of the domain. In other words, the values of the solution will be positioned on the nodes of the grid while the values of the diffusivity coefficient will be positioned in the intervals. Since we have a Dirichlet boundary condition, only the interior of the domain will be taken into consideration. A convergence analysis will also be performed by solving the PDE with several refinments of the mesh to determine if the rate of convergence agrees with the theoretical value of quadratic convergence. This work is joint with advisor Dr. Andrei Draganescu.

  5. Friday, December 17, 2010, 02:05-02:15
    Using Monte Carlo Method to Simulate Laser Propagation in Multilayer Tumor
    Yonghui Chen, Department of Mechanical Engineering
    Monte Carlo method simulations of photon propagation offer a flexible yet rigorous approach toward photon transport in turbid tissue. This method simulates the 'random walk' of photons in a medium that contains absorption and scattering. In my presentation I developed a Matlab program to simulate laser absorption, scattering, reflection and refraction in a multilayer tumor embedded in mice tissue. The result show that the energy of laser will mostly deposit in the target area as we expected. This work is joint with advisors Dr. Ronghui Ma and Dr. Liang Zhu.

  6. Friday, December 17, 2010, 02:20-02:30
    Simulations of a Model for Calcium Waves in a Heart Cell Using COMSOL Multiphysics
    David W. Trott, Department of Mathematics and Statistics
    Experiments have shown that calcium waves can occur from spontaneously generated calcium sparks in a heart cell. A model for this process is given by a system of reaction-diffusion equations. We have in the past developed a special-purpose C code with MPI for this model that successfully handles the high-resolution meshes of the desired three-dimensional domains. We will explore the basic framework of the model and investigate whether a general-purpose package such as COMSOL Multiphysics can be used. This work is joint with advisor Dr. Matthias Gobbert.

  7. Monday, December 13, 2010, 04:05-04:15
    ADI As Preconditioner for Krylov Subspace Methods
    Kyle Stern, Department of Mathematics and Statistics
    The alternating directions implicit (ADI) method is a classical iterative method for numerically solving linear systems arising from discretizations of partial differential equations. We propose to use ADI as a preconditioner for Krylov subspace methods for linear systems arising from the discretization of partial differential equations. This talk uses a finite difference approximation of an elliptic test problem as test problem for the computational experiments. The method is attractive because it allows highly efficient matrix-free implementations both in serial MATLAB and parallel C with MPI. This work is joint with advisor Dr. Matthias Gobbert.

  8. Wednesday, December 08, 2010, 04:05-04:15
    Modeling Tissue Necrosis Dependent Perfusion Coefficient during Magnetic Nanoparticle Hyperthermia
    Anilchandra Attaluri, Department of Mechanical Engineering
    Magnetic nanoparticle hyperthermia (MNH) is treatment procedure used to treat cancer in which heat is used to physically ablate the tumor. The Pennes bio-heat equation is solved using the finite element method on a spherical domain in one spatial dimension (in an interval) to obtain the temperature field. The goal of this work is to set up the model which can determine the total tissue damage due to the heat induced by MFH. The thermal damage is dependent on both the magnitude of change in temperature and the duration of the elevated temperature. From the energy point of view, the energy needed to elevate the tissue to a certain temperature during a time duration is directly proportional to both the temperature elevation and the treatment duration. Once the temperature field is obtained from the FEM solution of the bio-heat equation, an Arrhenius equation will be used to predict the tissue damage. This work is joint with advisor Dr. Liang Zhu.

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This page version 1.0, December 2010.