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# Numerical Methods for Partial Differential Equations

## CS555 :: Spring 2023

Are you interested in the numerical approximation of solutions to partial differential equations? Then this course is for you!

The course covers roughly three topics: the fundamentals of finite difference approximations, an introduction to finite volume schemes, and comprehensive look at finite element methods.

The course involves several assignments (usually bi-weekly) and a final project that we develop over the semester, culminating in a presentation. There is also a strong participation grade based on handouts and other in-class tasks. The tentative grade breakdown is:

• Homework 50
• Final Project 30
• Handouts/in-class activities 20 (we will be clear about these -- not all will be handed in)

This will be finalized in the first week of class.

The course assignments and examples in class will be in Python.

## Lectures

lecture zoom videos

Lecture Date Topic
01-18 About the course, classifying PDEs, survey of methods
01-23 Finite differencing for time dependent problems
01-25 Convergence theory, Stability
01-30 Stabilty, dispersion, and dissipation
02-01 Dispersion, and dissipation
02-06 Conservation laws, finite volume methods
02-08 Finite Volume Methods
02-13 Godunov Schemes
02-15 Higher resolution and slope limiting
02-20 Total Variation Diminishing
02-22 Multiple variables
02-27 Multiple dimensions
03-01 Discontinuous Galerkin for conservation laws
03-06 Discontinuous Galerkin for conservation laws, implementation
03-08 Projections
03-13
• Spring Break
03-15
• Spring Break
03-20 Weak forms, Ritz-Galerkin, existence and uniqueness
• Book Section 4.1, 4.2
03-22 Finite element interpolant, approximation property
• Book section 4.3
• project 0: Project selection + peer feedback
03-27 Finite element interpolant, approximation property
03-29 Finite element assembly
04-03 An overview of spaces
04-05 An outline of theory, Lax-Milgram, and more
04-10 An outline of theory, Lax-Milgram, and more
04-12 Finite elements and approximation properties
04-17 Back to DG
04-19 no class
04-24 Least-squares finite elements
• Book section 10.5
04-26 project: Presentations, day 0
• project 5: Project slides due
• Presentation 0.0 -- TBD
• Presentation 0.1 -- TBD
• Presentation 0.2 -- TBD
• Presentation 0.3 -- TBD
• Presentation 0.4 -- TBD
• Presentation 0.5 -- TBD
• Presentation 0.6 -- TBD
05-01 project: Presentations, day 1
• Presentation 1.0 -- TBD
• Presentation 1.1 -- TBD
• Presentation 1.2 -- TBD
• Presentation 1.3 -- TBD
• Presentation 1.4 -- TBD
• Presentation 1.5 -- TBD
• Presentation 1.6 -- TBD
05-03 project: Presentations, day 2
• Presentation 2.0 -- TBD
• Presentation 2.1 -- TBD
• Presentation 2.2 -- TBD
• Presentation 2.3 -- TBD
• Presentation 2.4 -- TBD
• Presentation 2.5 -- TBD
• project 5: Project reflections due

## Homework

Guidelines and files

## Final Presentations

The final presentation should focus on educating the audience with a clear statement of the problem (PDE) being looked at, a description of the method and its implementation, numerical results that highlight certain features of the study, and some reflective conclusions.

Your presentation should be 10 minutes long (!), with roughly 10 slides (depending on your content). Here is a rubric for the grading (equally weighted).

SP, Statement of problem

• Equations to be solved, including BCs and ICs if necessary. Connection to engineering/science problem if appropriate.

SA, Solution approach

• Discretization problem formulation, with justification. Why is this method of interest.

ID, Implementation details

• What details are needed to understand the implementation? What code(s) did you use?

R, Results

• What was the initial goal and did it succeed? What do you observe for this problem and discretization?

C, Conclusions

• What did you learn? What worked, what needs improvement, what are the next steps or future directions?
project step description points
project 0 idea 0
project 1 goals 1
project 2 outline 1
project 3 results 1
project 4 results 2
project 5 slides 20
project 6 reflections 5
total 30

# Computing

We will be using Python with the libraries numpy, scipy and matplotlib for assignments. No other languages are permitted.

# COVID and Attendance

While face coverings are not required in classrooms (current as of 01/17) we fully support your decision to wear one if you wish.

If you test positive for COVID, then you should not attend class.

If you have any cold-like symptoms or do not feel well, then you should not attend class, regardless of testing negative or positive for COVID.

In either case, your missed attendance due to illness will not impact your grade in the course and we will work with you to cover the material missed in class (via Zoom).