Math 621 - Numerical Methods for Partial Differential Equations
Spring 2015 - Matthias K. Gobbert
Presentations of the Class Projects

Tuesday, May 19, 2015, 01:00 p.m., SOND 109

  1. 01:05-01:20
    Efficient Simulation of the beta-alpha-delta (BAD) Model of Interactions in the Pancreatic Islet
    Sarah Swatski, Department of Mathematics and Statistics
    In 2012, 29.1 million Americans suffered from diabetes, a disease in which high blood glucose levels persist in the blood stream. The BAD model is a system of ODEs which represent the interactions between the beta, alpha and delta cells, found in the pancreas in clusters of cells called islets of Langerhans, which together control blood glucose levels. I implement this model in Matlab and compare the performance of the ODE solver after modifications are made. I recommend that a memory modified stiff ODE solver be used to most efficiently solve the problem.

  2. 01:25-01:40
    Implementation of the SDIRK Method for the Solution of the ODE System within a Method of Lines using Finite Element Discretization for Parabolic PDE
    Jonathan Graf, Department of Mathematics and Statistics
    The method of lines using finite elements results in a system of ODEs. In class, we examined the use of Matlab's ODE solvers for the solution of this system with particualr focus on ode15s. I will discuss the differences between an explicit RK method like ode45 and implicit RK method with a thorough discussion of the singly diagonally implicit Runge-Kutta (SDIRK) method. Convergence tables will be presented for linear and non-linear test problems.

  3. 01:45-02:00
    Numerical Methods for the Navier-Stokes Equations in 2D with Periodic Boundary Conditions
    Joshua Hudson, Department of Mathematics and Statistics
    The purpose of this project was to investigate the inherent difficulties in computing approximate solutions to the Navier-Stokes equations (NSE), by examining some approaches to computing approximate solutions based on a method of lines algorithm. In addition, other possible algorithms will be discussed which could not be implemented at the time. The complications imposed by the nonlinear term in the NSE was the main difficulty in the design and implementation of the algorithms studied. To simplify things, the equations were considered in the 2D setting with periodic boundary conditions, and with no forcing term. All computations were done in Matlab to allow for quick implementation of the algorithms, use of the differential equation software packages readily available, and the convenient plotting software.

  4. 02:05-02:20
    Finite Element Convergence Studies Using COMSOL 5.1
    Kourosh M. Kalayeh, Department of Mechanical Engineering
    The finite element method (FEM) is a well known numerical method for solving partial differential equations (PDEs). All numerical methods inherently have error in comparison to true solution of the PDE. Based on FEM theory, the appropriate norm of this error is bounded and it can be estimated by the mesh size. One standard method to get an idea about the sensibility of the numerical solution is to do convergence studies on the method. More precisely, compare the results obtained from two consecutive mesh refinements. This comparison, then can be quantified using theory of FEM by obtaining the convergence order. In this work we carry out convergence studies for time-dependent parabolic test problem (heat transfer equation) and investigate the effect of ODE solver on the behavior of the convergence of the method using commercial FEM software COMSOL 5.1.

  5. 02:25-02:40
    Finite Element Convergence Studies on Elliptic and Parabolic PDEs in MATLAB 8.5.0
    Preston Donovan, Department of Mathematics and Statistics
    The finite element method (FEM) is one of the most widely used numerical methods for solving PDEs. The Partial Differential Equation (PDE) toolbox in MATLAB 8.5.0, which uses the FEM, is applied to several elliptic and parabolic PDEs in 2D and 3D. The accuracy of the solutions is measured via qualitative and quantitative comparisons with the true solutions. Quantitative comparisons involve computing the norm of the error on progressively finer meshes and ensuring that the convergence order is consistent with FEM theory. We outline the process of performing a convergence study in MATLAB via the graphical user interface and the command line. Each of these processes presents its own challenges, which we explain and attempt to resolve.

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This page version 1.0, May 2015.