Course 3044 - Numerical Analysis II - Spring 2012

Official Information
Course Number:Mathematics 3044.001
Course Title:Numerical Analysis II
Times:TR 12:30-1:50
Places:BARTNB 205
Instructor: Benjamin Seibold
Instructor Office:518 Wachman Hall
Instructor Email: seibold(at) 
Instructor Office Hours:T 11:15-12:15, R 2:00-3:00
TA:Shimao Fan
TA Office:515 Wachman Hall
TA Email: tub00072(at) 
TA Office Hours:MW 2:00-3:30
Course Textbook: Brian Bradie, A Friendly Introduction to Numerical Analysis, Pearson Prentice Hall, 2006.
Official:Course Syllabus
Prerequisites:see Math Course Listing
Topics Covered: Adaptive quadrature, initial value problems of ordinary differential equations, Runge-Kutta methods, multistep methods, stiff problems, two-point boundary value problems, partial differential equations: Poisson equation, diffusion equation, advection equation.
Course Goals:Provide a sound working base in numerical methods, increase ability to apply proper mathematical tools to specific situations, introduce computing technology using MATLAB and apply it to problem solving, increase ability to work independently and formulate problem solving approaches, provide a set of experiences that can be utilized in other courses and beyond the classroom.
Attendance Policy:Attendance is required.
Course Grading:A(100-92), A-(91-90), B+(89-88), B(87-82), B-(81-80), C+(79-78), C(77-72), C-(71-70), D+(69-68), D(67-62), D-(61-60), F(below 60).
Grading:Homework 30%, Course project 30%, exams 40%; one midterm exam and one final exam.
Exam Dates:Midterm exam TBA; final exam Thursday 05/03/2012 from 10:30-12:30.
Homework:No late submits. No make-ups. Naked numbers are not acceptable. Solutions must include a short write-up describing the problem, your solution technique, and procedural details. To include a computer printout use the cut and paste method for placement of materials in your work. All things must be clearly labeled.
Computational Devices: You must ensure to have access to a computer, the internet, and the software package MATLAB to work on certain homework problems. MATLAB is available at various places on campus, for instance at the Tech Center.
Course Schedule
01/17/2012   Lec 1 Review and Overview
01/19/2011   Lec 2 Numerical differentiation and Newton-Cotes quadrature
01/24/2012   Lec 3 Adaptive quadrature (6.8)
01/26/2011   Lec 4 Initial Value Problems of ODE: ODE theory, examples
01/31/2012   Lec 5 Key numerical concepts (7.1), truncation errors, consistency, stability
02/02/2011   Lec 6 Euler's method (7.2), Taylor methods (7.3)
02/07/2012   Lec 7 Runge-Kutta methods (7.4)
02/09/2011   Lec 8 Implicit Runge-Kutta methods, Adams-Bashforth methods (7.5)
02/14/2012   Lec 9 Adams-Moulton methods (7.5), BFD methods (7.9), convergence and stability (7.6)
02/16/2011   Lec 10 Adaptive step size control (7.7)
02/21/2012   Lec 11 Absolute stability and stiff problems (7.9)
02/23/2011   Lec 12 Two-Point Boundary Value Problems: Finite difference method (8.1)
02/28/2012   Lec 13 Non-Dirichlet boundary conditions (8.2)
03/01/2011   Lec 14 Non-linear BVP (8.3), shooting methods (8.4 & 8.5)
03/13/2012   Lec 15 Elliptic Partial Differential Equations: Theory
03/15/2011   Lec 16 Poisson equation on a rectangle (9.1)
03/20/2012   Exam 1 Midterm Exam
03/22/2011   Lec 17 Irregular domains (9.5), comments on suitable linear solvers
03/27/2012   Lec 18 Parabolic Partial Differential Equations: Theory, method of lines (10.1)
03/29/2011   Lec 19 Temporal discretization (10.1)
04/03/2012   Lec 20 Large time steps and L-stability
04/05/2011   Lec 21 Von-Neumann stability analysis (10.2), more general parabolic PDE (10.3)
04/10/2012   Lec 22 Hyperbolic Partial Differential Equations: Examples, linear advection, upwind method (11.1)
04/12/2011   Lec 23 Modified equation
04/17/2012   Lec 24 Lax-Wendroff, Lax-Friedrichs
04/19/2011   Lec 25 Wave equation, staggered grids
04/24/2012   Lec 26 Course Project Presentations
04/26/2011   Lec 27  Course Project Presentations
05/03/2012 Final Exam
Matlab Programs
  • A second order accurate solver for the time-dependent heat equation; serves as a wrapper to test linear solvers: temple3043_heateqn.m
  • Simulation of an N-body problem in celestial mechanics. The example contains the sun, earth, jupiter, and the Voyager 1 probe, performing a swing-by at jupiter: temple3044_voyager.m
  • Perform numerical error analysis for the Poisson equation. A differentiable but oscillatory right hand side is considered: mit18336_poisson1d_error.m
  • Solves the Poisson equation in 1d, 2d, and 3d, and plots the sparsity patterns of the respective system matrices: temple3044_poisson.m
  • A finite difference solver for the 1D heat equation with time-dependent boundary conditions. Can easily modified to implicit time stepping schemes: temple3044_heateqn.m
  • Finite differences for the one-way wave equation, additionally plots the von Neumann growth factor: mit18086_fd_transport_growth.m
  • Finite differences for the wave equation, using a leapfrog method: mit18086_fd_waveeqn.m
Additional Course Materials
Homework Problem Sets
Course Projects
  • Zachary Ankuda: Heat exchange of a modern CPU heat sink
  • Matthew Berardi: Numerical methods of triangulating a geometry
  • Alexander Gonzalez: Modeling Cymatic waves using Matlab
  • Dylan Lexie: Tank wars in space
  • Kathryn Lund: Numerical solution of bound-state resonances on a perturbed cylinder
  • Elisheva Stern: Impact of driving behavior on fuel consumption
  • Zhou Ye: Numerical methods to solve initial value problem of Stochastic Differential Equations (SDE)
The midterm project reports are due 03/15/2012.