Numerical Solution of
Partial Differential Equations
Class meets Tuesdays, Thursdays, 3:30 PM to 4:45 PM in Phillips 301
Office hours: M-Th, 11:00 AM to 12:00 PM, (Phillips 307), e-mail appointment
The physical world is described by a small set of laws. One of the most useful mathematical statements of these physical laws is in the form of partial differential equations (PDE's) describing local changes in physical parameters. Though it is usually straightforward to write down the particular PDE's that correspond to a given problem, finding a solution is much more difficult. For the vast majority of real-world problems approximate techniques must be used. One of the most productive approaches is to use a numerical approximation of the PDE's of interest. These techniques are used in studying chemical reactions, ecological models, astrophysical phenomena, aircraft design, financial models and in many other application domains.
The development of numerical methods for PDE's and the ensuing theoretical development is within the province of applied mathematics. This course introduces the basic aspects of the discipline at the graduate level. At the end of the course the student should be able to understand the basic theory associated with solving each type of PDE encountered in practice, be adept at choosing methods appropriate for a specific application and be proficient in the computer solution of PDE's.
Grading shall be determined based on homework (HW, 48 points), supplementary reading (SR, 8 points), final project (FP, 16 points), midterm examination (ME, 14 points) and final examination (FE, 14 points). The number of points accumulated during classwork is mapped to a graduate grade as shown in the following table.
One of the tenets of graduate study is the ability to critically examine multiple sources in order to verify theories and to enhance understanding. Lecture notes for the course shall be provided online in Postscript and PDF formats. These are meant to document the progress of material presented in class. It is expected and required that the student supplement the lecture notes with reading from other sources.
The following texts serve as general background material. Students are encouraged to actively read from these texts material related to the current lecture. Homework and final examination questions from this material should be expected.
Finite Difference Schemes and Partial Differential Equations, John Strikwerda
Numerical Methods for Conservation Laws, Randall LeVeque
Finite Difference Schemes for Computational Fluid Dynamics, Phillip Colella & Gerry Puckett
Suggested reading for specific topics
Material specific to a certain topic shall be listed here as the course progresses. Generally the books or articles will be placed on reserve at the Mathematics Library in Phillips for the time period indicated
P.G. Garabedian, Partial Differential Equations, Chapter 1, The Method of Power Series. (17 pp.). On reserve: Aug. 22 - Aug. 29. Questions to ponder: (1) Consider the relevance and applicability of the methods used in the Cauchy-Kowalewski theorem to practical computations; (2) Try to find a biography of Sophia Kowalewski (also transcribed as Sonja Kovalevski) and place the Cauchy-Kowalewski theorem in historical context.
G. Dahlquist, Numerical Methods, Prentice-Hall, 1974, Sections 7.1 (Difference operators), 7.2 (Richardson extrapolation), Chapter 8 (Differential equations)
During the course, 6 homework assignments shall be given. Each homework shall contain 4 required topics and a bonus topic, each worth 2 points for a maximum of 10 points. The bonus topics are meant to allow for makeups of deductions on homework or examination grades. Homework assignments are given every second Tuesday and due two weeks thereafter. The purpose of the homework assignments is to achieve familiarity with the course material.
.ps = Postscript File, .pdf = Portable Document File, .html = Web page, .nb = Mathematica notebook .m = Matlab m file .f90=Fortran 90 source file
Supplementary reading assignments
Two reading assignments shall be given, each graded with 4 points. The purpose of the reading assignments is to introduce students to the tasks involved in research work. Points are awarded to this end as follows:
Students can freely choose when they want to carry out their reading assignment, but beware the temptation of putting things off until the last week of the semester. A rack with suitable papers is available in my office. Duplicate reading assignments are not allowed. A short report on each reading assignment must be prepared and turned in before the end of the semester.
It is essential that students acquire a basic familiarity with the process of developing and using scientific software. The homework assignments shall contain gradually more difficult problems that are intended to be solved using programs written by the students. The problems in the first homework assignments are small enough that they can solved using Matlab. Later problems are more computationally intensive and require writing programs in a compiler language such as C or Fortran.
Computer lab sessions
In order to aid proficiency in computer techniques a number of classes will be held in the computer lab (Phillips 324). Typically we will start from an initial file to be downloaded from this website and work through it.
Project ideas. Some typical project ideas are shown below. You can choose one of these or just browse through to get some inspiration for a theme of your own choosing.
Final projects will be defended in my office Phillips 307 by appointment up to December 10, 2003.
The midterm examination will be held Thursday, October 30 from 3:30 PM to 4:45 PM in Phillips 301. The syllabus comprises course material up to and including finite difference methods for the heat equation.
Questions similar to those students can expect on the midterm examination. (Some of these are from examinations covering more material than what is required for the midterm)
Final examination preparation
The final examination will be held Saturday, December 13 from 4:00 PM to 6:00 PM in Phillips 301. The syllabus comprises course material from the hyperbolic PDE's to the end of the course.
Questions similar to those students can expect on the final examination. .ps .pdf
Solutions: .ps .pdf