Numerical Methods II
Spring 2003, Mondays 5:00-7:00pm, WWH Room 101
Instructor: Olof B. Widlund
Office: WWH 712
Office Hours: Drop by any time, or send email or call for appointment
Regular homework assignments, including programming assignments,
using Matlab or any reasonable alternative.
It is important that you do the homework yourself, but when you get stuck,
I encourage you to consult with other students, or the math computer
consultant, or me, to get help when necessary. However, when you get help,
it's important to acknowledge it in writing. Passing off other people's
work as your own is called plagiarism and is not acceptable.
- January 27. Introductory remarks: Large problems with structure.
Polynomial interpolation, Newton's interpolation procedure,
Cubic Hermite interpolation, and cubic splines.
- February 3. Error bounds for Simpson's rule from error bounds for
polynomial interpolation. Adaptive Simpson. Existence and uniqueness
of initial value problems for ODEs. The convergence of Euler's method
and additional details on the structure of the error. Methods based
on Taylor series. Adams' methods.
- February 10. More about Adams' methods. Linear multistep methods.
The root condition. Dahlquist's first barrier theorem. Marginal stability
of the mid-point and Simpson schemes.
- February 17. Presidents' day. No class.
- February 24. Basics on stability diagrams and recipes on change of
step size in multistep methods. Backward differentiation schemes.
Predictor-corrector methods. Explicit Runge-Kutta methods.
- March 3. Implicit Runge-Kutta methods. Connection to collocation schemes
and Gaussian quadrature. Stiff problems and stability diagrams. Dahlquist's
theorem on A-stable methods. A(alpha)-stability and the BDF family. Pade
approximation and implicit Runge-Kutta methods. Error control.
- March 10. Poisson's equation. Dirichlet and Neumann boundary conditions.
Approximation of a two point boundary value problem with finite differences.
Discretizing Poisson's equation on general regions using finite differences.
The five-point formula and two nine-point schemes. Approximations next to
the boundary. The maximum principle and error bounds based on it. Poisson's
equation on a rectangle and the explicit form of the eigenvalues and
- March 17. Spring recess. No class.
- March 24. Poisson's equation on rectangular and other special regions.
Explicit eigen systems. Fast Poisson solvers based on the FFT. Special
block triangular matrices with commuting blocks. Direct solver of linear
systems with banded matrices. Band width for elliptic difference equations
in two and three dimensions.
- March 31. Bunemann's algorithm. Further comments on Gaussian elimination
and its limitations especially for three dimensional elliptic problems.
Row diagonally dominant matrices. Jacobi, Gauss-Seidel, and successive
overrelaxation methods. The convergence and relative quality of Jacobi
and Gauss-Seidel iterations for strictly row diagonally dominant matrices.
- April 7. Continuation of a discussion of classical iterative methods:
A sufficient condition for the convergence of SOR. The convergence of SOR
in the positive definite symmetric case. Property A and consistent ordering.
An optimal choice of the SOR parameter for consistently ordered matrices.
A short introduction to finite elements: Weak formulation of Poisson's
equation. Finite element triangulations. Continuous piece-wise linear functions.
- April 14. A finite element method for Poisson's equation. Triangulations,
element stiness matrices, sparsity of the stiffness elements, condition numbers
of the stiffness and mass matrices. A Hermite piecewise cubic continuous finite
element space and its description using barycentric coordinates.
- April 21. Error bounds for finite element methods. Approximation of
the heat equation with finite elements by semidiscretization. Solution of
the heat equation and its wellposedness in many norms. Max norm stability
of several simple difference approximations of the heat equation. The
use of Fourier transforms.
- April 28. Petrowskii's condition. An example of its limitations illustrated
by changing variables of a system. Choosing the correct transformation of
the wave equation to mke it possible to use the energy method. First order
systems of partial differential equations with symmetric coefficient matrices.
von Neumann's condition. Simple examples of difference approximations of
the advection equation.
- May 5. Conclusion of discussion of finite difference approximations of
A First Course in the Numerical Analysis of Differential Equations by Arieh Iserles. Cambridge University Press.
If you don't have a Sun workstation account, but you want one,
send me email.
Advice concerning computing:
See Professor Overton's
Numerical Methods I home page.
Don't Hesitate to Ask for Help
If you have questions, send me email, give me a call, or drop by my office.
Don't wait until it's too late!