# Numerical Methods II

### G22.2421, G63.2020 Spring 2003, Mondays 5:00-7:00pm, WWH Room 101 Instructor: Olof B. Widlund

• Coordinates
Office: WWH 712
Telephone: 998-3110
Office Hours: Drop by any time, or send email or call for appointment
Email: widlund@cims.nyu.edu

• Requirements
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.

• Homework
• Lectures
1. January 27. Introductory remarks: Large problems with structure. Polynomial interpolation, Newton's interpolation procedure, Cubic Hermite interpolation, and cubic splines.
2. 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.
3. 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.
4. February 17. Presidents' day. No class.
5. 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.
6. 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.
7. 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 eigenfunctions.
8. March 17. Spring recess. No class.
9. 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.
10. 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.
11. 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.
12. 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.
13. 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.
14. 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.
15. May 5. Conclusion of discussion of finite difference approximations of evolutionary equations.
• Required Text
A First Course in the Numerical Analysis of Differential Equations by Arieh Iserles. Cambridge University Press.

• Sun Account
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!