Course information for current students:
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.
14-1-05 (3)
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]
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?
1.7 Sources of error
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]
21-1-05 (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]
-.- Proof of convergence for the 5-pt-star FD method using a DMP [St 12.5,
MM 6.2]
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
2.4 General elliptic operator [HP 10.1, MM 6.3]
2.5 Non-uniform grid [HP 10.1]
28-2-05 (2.5)
-.- Example: derivation of 2nd-order non-uniform FD approximation to u'
2.6 Non-rectangular domains [CG 4.1.2, MM 6.4]
-.- Example: discretising Neumann BCs on a non-rectangular boundary [HP 10.2]
2.7 High order Finite Difference methods [Is 7.3, CG 5.2.1]
-.- Debugging and testing of PDE software -- first simple tests - FDMs
4-2-05 (2)
-.- Direct linear solvers, general, symmetric, banded, banded storage,
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
11-2-05 (3)
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]
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]
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
25-2-2005 midterm (1.5), lecture (1)
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)
4-3-2005 (2.5), some discussion on midterm
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]
3.12 Quadratic spline collocation in 2D
Other degree piecewise polynomials
11-3-2005 (2.5)
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
-.- Ellpack
18-3-2005 (2.5)
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]
4.2 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
4.7 Stability analysis
The von Neumann stability analysis method [Am 2.2]
The matrix stability analysis method [Am 2.5]
4.8 Consistency, Convergence and Stability [Am 2.7]
4.9 Convergence of the FD scheme for the heat equation
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
1-4-2005 (3)
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)
Tensor product formulation of PDE problems
-.1 Tensor products of matrices
-.2 Using tensor products to solve matrix problems arising from
finite difference or finite element methods
FFT solution of PDE problems
-.1 The Discrete Fourier Transform and the Fast Fourier Transform Algorithms
-.2 Solving one-dimensional BVPs using FFTs
-.3 Solving two-dimensional BVPs using FFTs
8-4-2005 (3)
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
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
The multigrid method