####
MAT 1062H:
Computational Methods for PDE

__ Professor__: Mary Pugh

__ Contact information__: mpugh@math. utoronto.ca

__ 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.

Syllabus

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