MAT 1062H: Computational Methods for PDE

Professor: Mary Pugh
Contact information: mpugh@math.
Office hours: by appointment
Office location: room 3141, Earth Sciences Centre, 22 Russell Street (To find my office, enter the building on the Northwest corner of Huron and Russell. Get to the third floor. Walk as far south as you can. Now walk as far west as you can.)

Meeting time and place: The class meets on Tuesdays 11:10am-12pm in BA1220 and Thursdays 10:10am-12pm in BA6183. The first lecture will be on on Tuesday January 8 and the last on Thursday April 10.

Goal: We'll study numerical methods for solving partial differential equations that commonly arise in physics and engineering. We will pay special attention to how numerical methods should be designed in a way that respects the mathematical structure of the equation.
  • Parabolic PDE: explicit and implicit discretizations in 1-d consistency, stability, and convergence in 1-d boundary conditions in 1-d multi-dimensional problems
  • Elliptic PDE: solution of sparse linear systems variational formulations and finite element methods
  • Hyperbolic PDE: CFL stabilty condition nonlinear conservation laws, shock capturing
  • Special topics: pseudospectral methods

    Why we care: Here are some disasters which could have been averted if only someone had been paying closer attention to their numerical analysis. :-)

    Prerequisites: You should be familiar with the material that would be taught in a serious undergraduate PDE course. Sample programs will be provided in matlab. If you know matlab, great! If you don't, you're expected to be sufficiently comfortable with computers that you can learn matlab on the fly. Which isn't actually hard at all, unless you hate computers.

    Recommended Reading: There are two books which provide background reading on numerical analysis, including numerical linear algebra, ODEs, finite difference methods, accuracy, and the like. Both are on reserve at the math/stat library on the 6th floor of Bahen. "An introduction to numerical analysis" by Kendall E. Atkinson is at the graduate level. "Elementary numerical analysis" by Kendall Atkinson and Weimin Han is at the undergraduate level. Also, I have asked that a book on numerical PDE be put on reserve at the physics library: "Finite difference schemes and partial differential equations" by John C. Strikwerda.


    Lecture Notes: Jan 8, 2008
    How to write up your homework
    Matlab Primer View a matlab primer. WARNING: Don't try to print the primer from acrobat, you'll get gibberish!
    A free online Matlab tutorial Note: google will turn up lots of hits on matlab and matlab itself has reasonable help pages.
    Make sure you can download and execute a file.
    Lecture Notes: Jan 10, 2008
    Diary from matlab demo in class, January 15
    On solving the heat equation using finite-difference methods.
    Lecture Notes: Jan 17, 2008
    First homework assignment, Due Tuesday January 29.
    Problem 2 here may help with problem 1 of your homework.
    Here's the demo from class on Jan 17. It needs the function find_spec.m
    Lecture Notes: Jan 22, 2008
    the matlab script that shows how I made and saved the plots from the Jan 22 lecture notes
    Lecture Notes: Jan 24, 2008
    Lecture Notes: Jan 29, 2008. Also, you can find the in-class demos here.
    Second homework assignment, due in class Thursday February 7, 2008
    Lecture Notes: Feb 5, 2008
    Lecture Notes: Feb 7, 2008. Also, you can find the in-class demos here.
    Third homework assignment. due in class Thursday February 28, 2008
    Lecture Notes: Feb 12, 2008
    Lecture Notes: Feb 14, 2008
    Convergence studies of schemes for initial data with different amounts of smoothness.
    Lecture Notes: Feb 26, 2008
    You can find the in-class demos for the advection equation here.
    Lecture Notes: Feb 28, 2008
    Lecture Notes: Mar 4, 2008
    Lecture Notes: Mar 6, 2008
    Fourth homework assignment. due in class Thursday March 20, 2008
    Lecture Notes: Mar 11, 2008
    Programs for conservation laws
    Lecture Notes: Mar 13, 2008
    Lecture Notes: Mar 18, 2008
    Some finite element programs
    Fifth homework assignment. due in class Tuesday April 1, 2008
    Lecture Notes: Mar 20, 2008
    Lecture Notes: Mar 27, 2008
    Lecture Notes: Apr 1, 2008
    Lecture Notes: Apr 3, 2008
    Lecture Notes: Apr 8, 2008
    Spectral programs for the heat equation, Burger's equation, and the cubic Schroedinger equation