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Numerical Methods for Partial Differential Equations (CS 555) Spring 2020

What	Where
Time/place	Wed/Fri 11:00am-12:15pm 1304 Siebel / Catalog
Class URL	https://bit.ly/numpde-s20
Live lecture	Participate » (requires password) · Backup: Twitch Stream
Class recordings	Watch » · Alternate, prior to spring break: Echo 360
Piazza	Discuss »
Calendar	View »
Office Hours	Participate » (Please use Chrome/Chromium to connect for now)

Homework

Homework 1 (due Feb 7)
Homework 2 (due Feb 26)
Homework 3 (due March 13)
Homework 4 (due April 15)
Homework 5 (due May 6)

Quizzes and Feedback Form

In-Class Quiz 1 (took place in class March 6)
No credit feedback form

Project

Project 1 (due April 24)
Project 2 (due May 15)
Project 2 (extra credit) (due May 15)

Why you should take this class

This course covers the basics of finite difference schemes, finite volume schemes, and finite element methods (majority). In addition, we'll cover some advanced topics such as discontinuous Galerkin and spectral methods.

One of the goals of this course is to build intuition for these methods. We'll be using Python and will be providing background for many of the computational and mathematical concepts in the course. As such, you do not need to be an expert in PDEs or in coding. But you should have a course in numerical analysis as your background (CS450 or equivalent), be comfortable with differential equations, and have some coding experience.

The course is divided in roughly two parts: hyperbolic and elliptic. This is of course a generalization, but it does allow us to focus on finite difference/finite volume methods for one part of the course and finite elements for another part. In addition to model problems we'll look at Stokes and other equations in order to develop a full understanding of the methods.

The course involves several homeworks (usually bi-weekly) and two projects: a midsemester project and a final project. There is also a participation grade based on quizzes.

The course homeworks and examples in class will be in Python. In particular, we'll use numpy and scipy.

Instructor

Andreas Kloeckner

(Instructor)

Email: andreask@illinois.edu

Office: 4318 Siebel

Teaching Assistant

Malachi Phillips

(Teaching Assistant)

Email: malachi2@illinois.edu

Office: 3401 Siebel

Course Outline

Note: the section headings in this tree are clickable to reveal more detail.

Introduction
Finite Difference Methods for Time-Dependent Problems
- 1D Advection
- Stability and Convergence
- Von Neumann Stability
- Dispersion and Dissipation
- A Glimpse of Parabolic PDEs
- Demo: Crank-Nicolson Scheme
- Demo: Dispersion and Dissipation
- Demo: Experimenting with Dispersion and Dissipation
- Demo: Methods for 1D Advection
- Demo: Truncation Error Analysis via sympy
- Demo: Waves and Stability
Finite Volume Methods for Hyperbolic Conservation Laws
- Theory of 1D Scalar Conservation Laws
- Numerical Methods for Conservation Laws
- Higher-Order Finite Volume
- Outlook: Systems and Multiple Dimensions
- Demo: Finite Volume Burgers
Finite Element Methods for Elliptic Problems
- tl;dr: Functional Analysis
- Back to Elliptic PDEs
- Galerkin Approximation
- Finite Elements: A 1D Cartoon
- Finite Elements in 2D
- Approximation Theory in Sobolev Spaces
- Saddle Point Problems, Stokes, and Mixed FEM
- Non-symmetric Bilinear Forms
- Demo: 2D FEM Using Firedrake
- Demo: 2D Stokes Using Firedrake
- Demo: Bad Discretizations for 2D Stokes
- Demo: Developing FEM in 1D
- Demo: Developing FEM in 2D
- Demo: Rates of Convergence
Discontinuous Galerkin Methods for Hyperbolic Problems
- Case Study: Maxwell's as a Conservation Law
- Evaluating Schemes for Advection
- Developing DG
- Fluxes and Stability
- Implementation Concerns
- Demo: Finding Numerical Fluxes for DG

Class scribbles (click to view)

CAUTION!

These scribbled PDFs are an unedited reflection of what we wrote during class. They need to be viewed in the context of the class discussion that led to them. See the lecture videos for that.

If you would like actual, self-contained class notes, look in the outline above.

These scribbles are provided here to provide a record of our class discussion, to be used in perhaps the following ways:

as a way to cross-check your own notes
to look up a formula that you know was shown in a certain class
to remind yourself of what exactly was covered on a given day

By continuing to read them, you acknowledge that these files are provided as supplementary material on an as-is basis.

Computing

We will be using Python with the libraries numpy, scipy and matplotlib for assignments. No other languages are permitted. Python has a very gentle learning curve, so you should feel at home even if you've never done any work in Python.

Virtual Machine Image

While you are free to install Python and Numpy on your own computer to do homework, the only supported way to do so is using the supplied virtual machine image.

Download Virtual Machine »

Books and Source Material

Draft Textbook

Once you sign in and complete your enrollment in RELATE, you will gain access to a draft textbook that was made available by Luke Olson.

Supplementary Text Books

Strikwerda, John C. Finite Difference Schemes and Partial Differential Equations. (available as an e-book via the UIUC library) Society for Industrial and Applied Mathematics, 2004. Second edition. DOI.