Math 221 Numerical Solution of Partial Differential Equations Part I 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 [Motivation] [Objectives] [Syllabus] [Grading] [Texts] [Assignments] [Computing] [Projects] [Midterm] [Final]   Motivation 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.   Course Objectives 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.   Syllabus Basic equations of mathematical physics, other application domains Minimal residual formulation Numerical methods for ODE's Linear advection diffusion equations Parabolic equations Hyperbolic equations Equations of mixed type Coupled PDE-ODE systems   Grading Policy 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. Clear Excellence H 87-100 points Entirely Satisfactory P 70-86 points Low Passing L 50-69 points Failed F 0-49 points   Course Texts 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. Lecture notes Lecture Date Topic Notes Lecture Date Topic Notes 1, 8/20 Overview of PDE's  .ps .pdf   10/10 Lab: advection Lab3 2, 8/22 ODE/PDE problems  .ps .pdf 15, 10/15 Modified equations  .ps .pdf 3, 8/27 Weighted residual  .ps .pdf   10/17 (Fall break - no class) 4, 8/29 ODE IVP  .ps .pdf 16, 10/22 Non-linear hyperbolic equations  .ps .pdf 5, 9/3 ODE num. methods  .ps .pdf 17, 10/24 Hyperbolic systems  .ps .pdf 6, 9/5 Euler method analysis  .ps .pdf 18, 10/29 Finite volume  .ps .pdf 7, 9/10 Stability, consistency  .ps .pdf 19, 10/31 Splitting methods  .ps .pdf 8, 9/12 Bnd. Loc., convergence  .ps .pdf   11/5 Lab: Spectral differentiation   9, 9/17 Fourier analysis  .ps .pdf 20, 11/7 Spectral methods  .ps .pdf 10, 9/19 1D Heat eq. FDM's  .ps .pdf 21, 11/12 Compact finite differences  .ps .pdf 11, 9/24 2D Heat eq.  .ps .pdf 22, 11/14 Finite element methods  .ps .pdf   9/26 Lab: Fourier space  Lab2  11/19 Lab: 2D Navier-Stokes equations   12, 10/1 Linear advection: characteristics  .ps .pdf 23, 11/21 Finite element formulations  .ps .pdf 13, 10/3 Basic advection methods  .ps .pdf 24, 11/26 Ritz formulation example  .ps .pdf 14, 10/8 Advection schemes: stability  .ps .pdf 35, 12/3 Project presentations   General bibliography 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)   Assignments Homework assignments 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. Date issued Date due Topic Homework Solution 8/26 9/9 Finite differences .ps .pdf 9/9 9/30 Numerical methods for ODE's .ps .pdf .ps .pdf 9/30 10/14 Heat equation algorithms .ps .pdf .ps .pdf 10/14 10/28 Linear hyperbolic equations .ps .pdf .ps .pdf 10/28 11/11 Non-linear hyperbolic equations .ps .pdf .ps .pdf .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: 1 point for reading the assigned text (I'll ask a quick question); 1 point for background research, i.e. placing the assigned text in the context of previous work for instance by reading the references from the assigned text; 1 point for forward research, i.e. investigating the ramifications of the assigned text using tools such as the Science Citation Index ; 1 point for working through some of the results from the assigned text, 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.   Computational work 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. 9/4 Introduction to Mathematica, Finite differences lab1.nb  9/25 ODE stability lab2.nb 10/9 Real, Fourier space behavior of common PDE's lab3.m 10/23 lab4.m 11/20 Spectral method for Navier-Stokes equations specaccuracy.m lab5.m wavenumber.m Final project topics 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. 1. Antisound 2. Optimal placement of effluent pipes 3. Sound barrier 4. Target identification through sound 5. Target identification thorugh heat 6. Beach erosion   Final projects will be defended in my office Phillips 307 by appointment up to December 10, 2003.   Midterm examination preparation 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) 1.pdf 2.pdf   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