NUMERICAL METHODS II (CSCI 2421 / MATH 2020)
Fall 2024
Instructor
Mike O'Neil (oneil@cims.nyu.edu)
Lecture
Mon & Wed 11:00am - 12:15pm, WWH 1302
Office hours
Mon 3:00pm-4:00pm, WWH 1119
Wed 2:00pm-3:00pm, WWH 1119
(or by appointment)
Description
This course is focused on numerical methods for solving ordinary and partial differential equations, and will include topics such as: numerical approximation theory, orthogonal polynomials, the Fast Fourier Transform, finite differences, spectral methods, 2-point boundary value problems, elliptic PDEs and integral equations, high-order quadrature techniques, and fast structured matrix computations. View the syllabus for more detailed information.
Mike O'Neil (oneil@cims.nyu.edu)
Lecture
Mon & Wed 11:00am - 12:15pm, WWH 1302
Office hours
Mon 3:00pm-4:00pm, WWH 1119
Wed 2:00pm-3:00pm, WWH 1119
(or by appointment)
Description
This course is focused on numerical methods for solving ordinary and partial differential equations, and will include topics such as: numerical approximation theory, orthogonal polynomials, the Fast Fourier Transform, finite differences, spectral methods, 2-point boundary value problems, elliptic PDEs and integral equations, high-order quadrature techniques, and fast structured matrix computations. View the syllabus for more detailed information.
Materials
Much of the course material will be drawn from:
- L. Nick Trefethen, Spectral Methods in MATLAB, SIAM, 2000.
- Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007.
The above links point to the electronic versions of these books available through NYU Libraries. See the syllabus for additional reference textbooks. Some more advanced material will be based on journal papers and other textbooks.
Grading
The overall course grade will be determined from a final numerical weighted average. The following breakdown will be used to compute an overall numerical grade:
- 70% Homework (roughly bi-weekly, lowest grade dropped)
- 30% Take home final, or Final project + presentation (details to follow)
Announcements
- Grades and a link to Gradescope for homework submissions has been setup on Brightspace here.
Much of the course material will be drawn from:
- L. Nick Trefethen, Spectral Methods in MATLAB, SIAM, 2000.
- Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007.
The above links point to the electronic versions of these books available through NYU Libraries. See the syllabus for additional reference textbooks. Some more advanced material will be based on journal papers and other textbooks.
Grading
The overall course grade will be determined from a final numerical weighted average. The following breakdown will be used to compute an overall numerical grade:
- 70% Homework (roughly bi-weekly, lowest grade dropped)
- 30% Take home final, or Final project + presentation (details to follow)
Announcements
- Grades and a link to Gradescope for homework submissions has been setup on Brightspace here.
Schedule
Below is an updated list of discussion topics along with any documents that were distributed, relevant reference source, etc.| Date | Topics | Materials |
|---|---|---|
| Jan 22 | Course overview, computing environments, discretizing derivatives |
Trefethen, Chapter 1 LeVeque, Appendix A |
| Jan 27 | Function approximation, spectral differentiation |
Trefethen, Ch 1, 2, 3 Briggs & Henson, Ch 1, 2, 3, 6 S. Johnson, Notes on FFT-based differentiation, 2011 |
| Jan 29 | Spectral differention, bandlimited interpolation |
Trefethen, Ch 3 S. Johnson, Notes on FFT-based differentiation, 2011 |
| Feb 3 | Fast Fourier Transform, convergence of Fourier series |
Katznelson, Ch II Briggs & Henson, Ch 10 Extension of Chebfun to Periodic Functions Trefethen, Ch 4 |
| Feb 5 | More convergence, Chebyshev differentiation |
Extension of Chebfun to Periodic Functions Trefethen, Ch 4, 5, 6 |
| Feb 10 |
Chebyshev differentiation, spectral methods for boundary value problems |
Trefethen, Ch 6, 7, 8, 13 |
| Feb 12 |
Pseudospectral vs Galerkin, discrete convolutions and FFTs |
|
| Feb 18 | Integral equations and spectral integration | |
| Feb 19 | Intro to finite differences: errors, stability | |
| Feb 24 | Finite differences: consistency, stability, convergence. | |
| Feb 26 | Finite differences: special cases and higher dimensional elliptic equations | |
| Mar 3 | Iterative solvers: Conjugate gradients, Krylov subspaces | |
| Mar 5 | Iterative solvers: GMRES | |
| Mar 10 | Boundary integral equations |
|
| Mar 12 | Nystrom discretization |
|
| Mar 17 |
Nystrom discretization Initial value problems |
|
| Mar 19 | Initial value problems: multistage and multistep | |
| Mar 31 | Initial value problems: Zero-stability and absolute stability | |
| Apr 2 | Initial value problems: Stability and stiff equations | |
| Apr 7 | Parabolic PDEs | |
| Apr 9 | Parabolic PDEs: Convergence, Von Neumann analysis | |
| Apr 14 | Parabolic PDEs: Von Neumann analysis, multidimensional problems | |
| Apr 16 | Nested dissection |