NUMERICAL METHODS II (CSCI 2421 / MATH 2020)
Spring 2026
Instructor
Mike O'Neil (oneil@cims.nyu.edu)
Lecture
Mon & Wed 11:00am - 12:15pm, WWH 1302
Office hours
Mon 2:30PM-3:30PM, WWH 1119
Tue 2:00PM-3:00PM, WWH 1119
Fri 11:00AM-12:00PM, WWH 705 (TA)
(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 2:30PM-3:30PM, WWH 1119
Tue 2:00PM-3:00PM, WWH 1119
Fri 11:00AM-12:00PM, WWH 705 (TA)
(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:
- 60% Homework (roughly bi-weekly, lowest grade dropped)
- 40% Take home final, or Final project + presentation (details to follow)
Announcements
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:
- 60% Homework (roughly bi-weekly, lowest grade dropped)
- 40% Take home final, or Final project + presentation (details to follow)
Announcements
Schedule
Below is an updated list of discussion topics along with any documents that were distributed, relevant reference source, etc.| Date | Topics | Materials |
|---|---|---|
| Jan 21 | Course overview, computing environments, discretizing derivatives |
Trefethen, Ch 1 LeVeque, Appendix A |
| Jan 26 | Basic Fourier analysis | Trefethen, Ch 1, 2, 3 |
| Jan 28 | Fourier transform, semi-discrete Fourier transform, Fourier series |
Trefethen, Ch 1, 2, 3 |
| Feb 2 | Function approximation, spectral differentiation |
Trefethen, Ch 1, 2, 3 Briggs & Henson, Ch 1, 2, 3, 6 S. Johnson, Notes on FFT-based differentiation, 2011 |
| Feb 4 | Spectral differention, bandlimited interpolation, FFT |
Trefethen, Ch 3 S. Johnson, Notes on FFT-based differentiation, 2011 Briggs & Henson, Ch 10 |
| Feb 9 | Fast Fourier Transform, convergence of Fourier series |
Katznelson, Ch II Briggs & Henson, Ch 10 Extension of Chebfun to Periodic Functions Trefethen, Ch 4 The Exponentially Convergent Trapezoidal Rule |
| Feb 11 | More convergence, Chebyshev differentiation |
Extension of Chebfun to Periodic Functions Trefethen, Ch 4, 5, 6 |
| Feb 17 |
Spectral methods for boundary value problems, Pseudospectral vs Galerkin, discrete convolutions and FFTs |
|
| Feb 18 | Integral equations and spectral integration |
|
| Feb 23 | Intro to finite differences: errors, stability |
|
| Feb 25 | Finite differences: consistency, stability, convergence. |
|