CSC446-2310S Computational Methods for Partial Differential Equations

Spring 2009

Course information for current students:

9-4-2009: In assignment 4, q1, the PDE is u_t = u_xx + u_yy (and not u_t = u_xx).
1-4-2009: If you need to pick up a hard-copy of assignment 4, you can come to BA 4226 Thursday 2-3 p.m. or Friday 2-3 p.m. to get one.
27-3-2009: Assignment 4 is now available towards the bottom of this page. I will bring copies to the lecture on Wednesday. Note that the due date is Tuesday, April 14, 2009, 2 p.m.
26-3-2009: A note on assignment 3: the interior collocation method does not accept any parameters; it works only with Hermite cubic C^1 pps, that satisfy the boundary conditions by construction.
12-3-2009: As mentioned in class, there will be no tutorial on Friday 13 March 2009. There will be a tutorial on Friday 20 March 2009. Please bring any question you have about the assignment. This (20-3-2009) will most likely be the last Friday we meet. We will keep only the Wednesday meetings after that.
22-2-2009: Assignment 1 is marked and available for pickup. My office hours for this Tuesday 24 Feb 2009 are extended from 4:10 p.m to 5:30 p.m. (always in BA 4226).
18-2-2009: Reminder: The term test is Wednesday, February 25, 1:10 p.m. and will be about one and a half hour long.
The material covered includes all things from the start of the course till the end of the BVP - Finite Difference Methods section. It does not include BVP - Finite Element Methods. So, the lectures we did till 6-2-09 (see below) are included in the exam.
Examples of questions for an exam:
You may want to see a practice question Also, question 1a, b, c, and question 2 of assignment 1 are good examples.
By Tuesday, February 24, I will have assignment 1 marked, and you can come during office hours to pick it up.
21-1-2009: We will have a regular class on Friday 23-1-2009. We will not have a class (nor tutorial) on Friday 30-1-2009.
7-1-2009: It came to my attention that there is yet another numerical PDE course offered at UofT. It seems to have a stronger mathematical and less computational flavour than CSC446-2310. See MAT 1062 -- Introductory Numerical Methods for PDEs for more details. Depending on what flavour you are looking for, you may be interested in auditing it or enrolling in it.
3-1-2009: The Friday 2-3 slot will be used primarily for lectures.
Lectures start Wednesday 7 January 2009, 1-3, BA B024.

Bulletin board for CSC446 and CSC2310 Spring 2009

Below I give some references related to the material taught in class.

Material covered in CSC446-2310 so far with references in brackets.
Abbreviations:
Am = Ames
CG = Celia and Gray
HP = Hall and Porsching
Is = Iserles
MM = Morton and Mayers
Pr = Prenter
St = Strikwerda
BGO= Becker, Garey and Oden
RB = Rice and Boisvert
See course handout for complete description of references.

7-1-09 (2)
1    Introduction
1.0  PDEs [CG 1.2]
1.1  Classes of problems and PDEs [Am 1.2,3, CG 1.2]
1.2  Some definitions [CG 1.2]
1.3  Boundary conditions [HP 2.3, 10.1]
9-1-09 (1)
1.4  Classes of numerical methods for PDEs [Am 1.4-8, CG 1.2-3]
1.5  Analysis of numerical methods for PDEs [Am 1.11]
1.6  How to measure the error in the approximation and
     evaluate a numerical method for a PDE?
     (start)
14-1-09 (2)
1.6  How to measure the error in the approximation and
     evaluate a numerical method for a PDE?
     (end)
1.7  Sources of error
1.8  How to numerically estimate the order of convergence of a PDE discr. meth.

2    Boundary Value Problems (BVPs) -- Finite Difference Methods
2.0  FD approximations to derivatives [Is 7.1, HP 10.1, CG 2.1-4,7, Am 1.5-6,
     MM 2.5]
-.-  Example: derivation of 2nd-order uniform & centered FD approximation to u''
2.1  A FD method for a model 1D BVP [CG 2.5]
2.2  A FD method for a model 2D BVP [Is 7.2, HP 10.1,5, CG 2.8, Am 3.1, MM 6.1]
16-1-09 (1)
2.3  Other than Dirichlet boundary conditions [HP 10.2, Am 3.1, MM 6.4]
-.-  Example: discretizing Neumann conditions for a 1D model BVP
     First order approximation
     Centered second order FD approximation
     One-sided second order FD approximation
-.-  Non-centered and one-sided FD formulae
     Example: one-sided second order approximation to the first derivative
-.-  Example: discretizing periodic conditions for a 1D model BVP
21-1-09 (2)
-.-  Proof of convergence for the 5-pt-star FD method using a DMP [St 12.5,
     MM 6.2]
2.4  General elliptic operator [HP 10.1, MM 6.3]
2.5  Non-uniform grid [HP 10.1]
2.6  Non-rectangular domains [CG 4.1.2, MM 6.4]

-.-  Example: derivation of 2nd-order non-uniform FD approximation to u'
-.-  Example: discretising Neumann BCs on a non-rectangular boundary [HP 10.2]
-.-  Example: derivation of 4th-order uniform FD approximation to u''
2.7  High order Finite Difference methods [Is 7.3, CG 5.2.1]
     (start)
28-1-09 (2)
2.7  High order Finite Difference methods [Is 7.3, CG 5.2.1]
-.-  Debugging and testing of PDE software -- first simple tests - FDMs
-.-  Direct linear solvers, general, symmetric, banded, banded storage,
4-2-09 (2)
     application to the 5-point-star matrix, fill-in, pivoting [Is 9.1, A1]
     Sparse solvers, sparse storage schemes
-.-  Norms (vector, matrix, function - discrete and continuous) [Is A2, CG A2]
     Approximating continuous norms - studying convergence
2.8  Adaptive mesh generation
6-2-09 (1)
2.8  Adaptive mesh generation (end)

3    Boundary Value Problems (BVPs) -- Finite Element Methods
3.1  FE approximating spaces and basis functions [CG 3.1.1, Pr 4.2-3, BGO 2.6]
11-2-09 (2)
     Piecewise polynomials and splines
     Constant pps, linear splines (C^0), quadratic splines (C^1),
     quadratic pps C^0, cubic splines (C^2), cubic Hermite pps C^1.
3.2  Multi-dimensional FE approximating spaces and basis functions
3.3  Adjusting FE approximating spaces and basis functions to BCs
3.4  Weighted residual methods -- Finite Element Methods [CG 3.1.2, Am 1.7]
13-2-09 (1)
-.-  Ellpack
Spring break
25-2-09 (2) -- midterm
27-2-09 (1)
3.5  A Galerkin method for a 1D BVP [Is 8.1, CG 3.3.1, 3.7, *BGO* 1.2-8, 2.3-4]
     Properties of stiffness matrix: symmetry, positive definiteness,
     bandedness, summability
4-3-09 (2)
some discussion on midterm
3.6  General operator and boundary conditions [CG 3.3.4, BGO 2.7]
3.7  A Galerkin method for a 2D BVP [CG 3.5.1-2]
3.8  Other degree piecewise polynomials (Galerkin)
6-3-09 (1)
3.9  Quadratic spline interpolation in 1D
     Other degree piecewise polynomials [related: Pr 4.2]
3.10 Quadratic spline interpolation in 2D
3.11 Quadratic spline collocation in 1D [related: Pr 8]

11-3-09 (2)
3.12 Quadratic spline collocation in 2D
     Other degree piecewise polynomials
3.13 Interpolation and collocation with certain types of BCs
     Quadratic spline collocation with homogeneous Dirichlet BCs
3.14 Error bounds for FEM approximations
3.15 Optimal spline collocation methods

18-3-09 (2)
4.   Initial Value Problems
  A  Parabolic Problems
4.1  An explicit one-step FD method for the model parabolic IVP (forward differences) [Am 2.1]
4.9  Convergence of the FD scheme for the heat equation
4.2  Attempts to improve the order of convergence or the stability conditions [Am 2.1, 2.4, 2.6]
4.3  Implicit versus explicit methods [Am 2.3]
4.4  An implicit one-step FD method for the model parabolic IVP (backward differences) [Am 2.3]
4.5  The Crank-Nicolson method [Am 2.3]
4.6  Computational complexity issues

20-3-09 (1)
4.7  Stability analysis
     The von Neumann stability analysis method [Am 2.2]
     The matrix stability analysis method [Am 2.5]
4.10 Von Neumann stability analysis of the FD scheme for the heat equation
4.11 Matrix stability analysis of the FD scheme for the heat equation

25-3-09 (2)
4.8  Consistency, Convergence and Stability [Am 2.7]
[4.9  Convergence of the FD scheme for the heat equation]
4.12  The method of lines
  B  Hyperbolic Problems
4.13 An explicit FD method for the model hyperbolic IVP (centered differences)
4.14 Implicit FD methods for the model hyperbolic IVP (Crank-Nicolson)
4.15 Alternating Direction Implicit (ADI) methods

1-4-09 (2)
5    Iterative methods for solving linear systems
5.1  General
5.2  Richardson, Jacobi, Gauss-Seidel, SOR and SSOR
5.3  Convergence of iterative linear solvers
5.4  The conjugate gradient method
5.5  Preconditioning

8-4-09 (2)
     Incomplete factorisation preconditioning
     Block diagonal preconditioning
5.6  The preconditioned conjugate gradient method
5.7  Block iterative methods
     The block Jacobi method for the 5-point-star matrix
     The block Gauss-Seidel and related methods for the 5-point-star matrix
5.8  Computational and memory requirements of iterative methods
5.9  Implementation and performance of iterative methods for the model problem

6.   The multgrid method
To be done next:

Files

Access to the following files requires that you use your CDF username and the last 5 digits of your student id.
Outline of course
Introduction
FDMs and BVPs
Debugging and testing DE software
Direct linear solvers and PDEs
Inner products and norms
ELLPACK
FEMs and BVPs
Galerkin
Interpolation and Collocation
Assignment 1
Assignment 2
Sample ELLPACK file 1 Sample ELLPACK file 2
Assignment 3
Files for a3: ReadMe, main21.m, setgrid.m, bsplex.m, bsplvd.m, bsplvn.m, coefpde.m, sc21.m, sc21u.m, eval21.m, error1.m, convergence.m, pde.m, bc.m, truevd.m, sptrid.m, intrvl.m
Parabolic and hyperbolic PDEs
Assignment 4 , least squares fit (if you need it)
Iterative Solvers