https://github.com/carlosmunozmoncayo/amcs-394d-2023

Materials for a course on nonlinear waves

https://github.com/carlosmunozmoncayo/amcs-394d-2023

Repository

Materials for a course on nonlinear waves

https://github.com/carlosmunozmoncayo/AMCS-394D-2023/blob/main/

# AMCS 394D: Nonlinear Waves

## Description

This course will cover mathematical models for a wide range of nonlinear wave phenomena from both a theoretical and a numerical perspective.   Nonlinear wave phenomena can broadly be divided into shock waves and dispersive waves; in the course we will cover both areas as well as some ways in which the two realms intersect.  Some of the topics that we will discuss include:

- Dispersion relations and the effects of disperson
- Hyperbolic PDEs: Nonlinearity, weak solutions, and shock waves
- Solitons and some basics of integrable systems
- Godunov-type finite volume methods for shock-dominated problems
- Pseudospectral methods for dispersive problems
- Derivation of various water wave models: Korteweg-de Vries, Boussinesq, BBM, Saint-Venant
- Dispersive shock waves
- Effectively dispersive hyperbolic systems

Later in the course we will go into some areas of current research, depending on the interests of the students.

By the end of the course, students will have an understanding of a range of nonlinear wave models and techniques for understanding them and computing their solutions.

## References:
- G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1974): https://onlinelibrary.wiley.com/doi/book/10.1002/9781118032954
- B. Deconinck, Nonlinear Waves course notes (2018): https://www.dropbox.com/s/48tyxnhtqczew28/NLWbook.pdf?dl=0
- R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems: https://www.cambridge.org/core/books/finite-volume-methods-for-hyperbolic-problems/97D5D1ACB1926DA1D4D52EAD6909E2B9
- D. I. Ketcheson, R. J. LeVeque, and M. J. del Razo, Riemann Problems and Jupyter Solutions, http://www.clawpack.org/riemann_book/
- [PseudospectralPython short course](https://github.com/ketch/PseudoSpectralPython)

## Evaluation

### Exercises (50%)
Exercises will be assigned regularly (usually one or two at the end of each lecture).  Typically solutions will be presented at the next class meeting or one week after being assigned.  Your grade for this portion depends on both (a) presenting solutions; and (b) actively participating in discussions of other students solutions.

You are encouraged to work with other students on the exercises.  Exercises will involve both programming and analysis.

### Course project (50%)
The remainder of your grade for the course will be based on completion of a research project.  For the project, you will lead a discussion of a paper on some topic related to the course.  Some [suggested papers](https://github.com/ketch/AMCS-394D-2023/blob/main/discussion_papers.md) are available, or you may choose your own (with the instructor's approval).  All class members will read the paper in advance of the discussion.  Your objective is to have a full understanding of the paper so that you can teach its contents to the others.  You may also wish to prepare computational examples or other aids in advance.  These discussions will take place on the last 5 Thursdays of the course, starting April 11th.

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