Finite Difference Methods for Partial Differential Equations
Professor: Paul J. Atzberger
206C Spring 2010, Meeting in Girv. 2112
TR 9:30am - 10:45am

Attach:classImage.jpg Δ

Welcome to the class website for the course Finite Difference Methods for Partial Differential Equations. The class will focus on numerical approximation of partial differential equations using finite difference approaches. The class will cover both mathematical foundations and practical aspects of effective implementation of such numerical methods. Many examples will also be discussed drawn from problems arising in the sciences, engineering, and finance. For more details see the syllabus and the topics listed below.

Please be sure to read the prerequisites and grading policies for the class.

Selection of Topics

• Introduction to Finite Difference Approximation.
• Hyperbolic, Parabolic, Elliptic Classification of Second Order PDEs.
• Finite Difference Methods for PDEs from each of these classes (Hyperbolic, Parabolic, Elliptic).
• Convergence and Consistency.
• Stability.
• Courant-Friedrichs-Lewy Condition.
• Lax-Richtmyer Equivalence Theorem.
• Fourier Analysis / Von Neumann Analysis.
• Order of Accuracy of Methods.
• Solving Sparse Linear Systems.
• Fast Poisson Solvers via FFT and Multigrid.

Prerequisites:

Ordinary Differential Equations, Partial Differential Equations, and Linear Algebra.

Grading:

The grade for the class will be based on the homework assignments (see policy below), midterm exam, and final project as follows:

Homework Assignments 30%
Midterm Exam 30%
Final Project 40%

Homework Policy:

Assignments will be made weekly and posted on the class website. Prompt submission of the homework assignments is required. While no late homework submissions will be accepted, one missed assignment will be allowed without penalty. While it is permissible for you to discuss materials with classmates, the submitted homework must be your own work. The assignments will consist of a combination of analytic problems and numerical calculations. Basic programming in Matlab/Octave may also be required for some assignments.

Exams:

A midterm exam will be given in the class on Tuesday, May 25th.

Midterm Outline

Midterm Solutions Δ

Final projects will be announced toward the end of the quarter.

Supplemental Class Notes:

GNU Octave Software and Documentation
Octave Software (Binaries from SourceForge.net)
GNU Octave Tutorial
GNU Octave Links (Tutorials and other Information)
Ubuntu (Linux Operating System) [Maybe useful to install dual-boot for running Octave]

Matlab Software and Documentation

Explosion of the Ariane 5 Rocket (Consequence of Faulty Numerics)
The Sinking of the Sleipner A Offshore Platform (Consequence of Faulty Numerics)

Class Annoucements:

- The grader for the class is Daniel Salazar (dsalazar@math.ucsb.edu). His office is South Hall 6431 J.

- The TA will post HW3 in an envelope outside his office in South Hall 6431 for pick-up if you would like it before the midterm exam.

- Midterm Outline

- Midterm Solutions Δ

Homework Assignments:

Turn all homeworks into the graders mailbox Daniel Salazar in South Hall 6th Floor by 5pm on the due date. Graded homeworks will be returned in class.

HW1: (Due Tue, April 13) (second edition see [PDF] Δ) (worked out example problem for the method of characteristics [PDF] Δ) 1.1: 1, 3, 4, 7. 1.3: 1abcd, 2, 3.

HW2: (Due Fri, April 30) 2.1.9, 2.1.10, 2.1.12, 2.2.1, 2.2.5, 2.3.1.

HW3: (Due Fri, May 14) Problem 1 from lecture. 3.1.1, 3.1.2, 3.1.6, 8.1.1, 8.1.2, 8.1.4, 8.1.11, 8.2.1.

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