$\parbox{5in}{ Algorithms and Numerical Solutions of Differential Equations \\ (Math 428 sec. 010)}$ Spring 2002
GOR 315 - Tuesday & Thursday 1530-1645
©2002 L. F. Rossi All rights reserved.

Prof. L. F. Rossi
Office: Ewing 524
Telephone: x1880
Email: rossi@math.udel.edu
WWW: http://math.udel.edu/$\sim$rossi
Computer project number: 2166

Course description: This course focuses on the numerical integration of equations. Scientific computation can be studied abstractly, and it can be practiced by professionals to solve problems. This course emphasizes both in equal parts. There is a rich mathematical theory underlying the application of numerical algorithms to various problems, and there is a lot to be learned by implementing numerical algorithms and studying their performance on specific problems.

Prerequisites: Math426 or CISC410 is a prerequisite for this course.

Your objectives:

I hope everyone will hone their mathematical abilities to reason quantitatively, logically, confidently and eventually correctly. In this course, everyone will master the following skills:

• Understand and use mathematical basics such as Taylor's theorem to study algorithms.

• Analyze numerical quadrature schemes. Connect these ideas to restriction and prolongation operators and interpolating functions.

• Implement quadrature schemes for general functions.

• Understand and implement Richardson extrapolation for numerical quadrature and other problems.

• Understand and implement simple, single and multi-step integration schemes.

• Use canned subroutines (from Matlab for instance) judiciously to solve more complex problems.

• Understand the accuracy and stability of ordinary differential equation solvers.

• Use ordinary differential equation solvers to resolve boundary value problems using shooting or finite difference methods.

• Analyze and use stiff solvers for stiff systems of differential equations.

• Analyze and use methods to solve simple parabolic partial differential equations.

Your resources: All of the following will help you achieve your objectives.

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Time: The best way to learn mathematics is to spend time with it. It is your responsibility to come to class each day, and it is unlikely that you will pass if you miss more than a day or two, regardless of the reason. Attendance is crucial to success in this course.

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Textbook: Numerical Analysis $2^{\rm nd}$ ed. by Kincaid & Cheney. I will not follow the book religiously, but the book will provide a good reference for most of the material.

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Office hours: Monday 0900-1000, Thursday 0900-1000, Friday 1030-1130 or by appointment. This is time that I have set aside especially for you. Office hours are one of the most valuable and least used resources at the University, and I hope you will take advantage of them. I want to help you learn this material, so do not be shy about seeing me outside of class. Your exams will only be distributed during office hours. If you need to see me at a time other than an office hour, feel free to ``drop in'' or make an appointment.

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Computers: You may use any computer and any programming language for this course. Since some assignments will make use of Matlab's differential equation solvers, I strongly encourage everyone to use Matlab for everything, but the choice is yours.

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Your classmates: Math is not a competitive sport. There are many obvious reasons to work together. Even if you end up helping others most of the time, teaching is one of the best ways to gain a deeper understanding of a subject.

Grading policy: Your grade is determined solely by your understanding of mathematics and your ability to communicate this knowledge to me on exams and other assignments.

Homework	20%
Projects (15% each)	45%
Exams (8% each)	24%
Final exam	11%

Final letter grades will be assigned strictly based on the following percentages of your total point score:
100 $\leftarrow$ A $\rightarrow$ 93 $\leftarrow$ A- $\rightarrow$ 90 $\leftarrow$ B+ $\rightarrow$ 87 $\leftarrow$ B $\rightarrow$ 83 $\leftarrow$ B- $\rightarrow$ 80 $\leftarrow$ C+ $\rightarrow$ 77 $\leftarrow$ C
C $\rightarrow$ 73 $\leftarrow$ C- $\rightarrow$ 70 $\leftarrow$ D+ $\rightarrow$ 67 $\leftarrow$ D $\rightarrow$ 63 $\leftarrow$ D- $\rightarrow$ 60 $\leftarrow$ F $\rightarrow$ 0



I reserve the right to adjust this scale to improve grades if the course material proves to be unreasonably demanding.

Exams: All exams will occur in class on the days listed on the syllabus. There are no makeup exams without prior notification and a valid, documented reason. Graded exams will be handed back in person in office hours only.

Problem sets: I will drop your lowest three homework scores during the semester. I do not accept late homework, so do not squander these three assignments. You might be sick sometime and not be able to do your homework on time.

Projects: Projects are larger scale problems that require you to implement computational algorithms to solve a problem.

Student conduct: To provide the best learning environment for all my students, I expect all my students to conduct all their scholarly activities with honesty and integrity. Students should note that in certain situations doing nothing can be dishonest. Though I hope there will never be a need to address academic dishonesty, I will strongly enforce all provisions noted in the Academic Regulations for Undergraduates. See The University of Delaware Undergraduate and Graduate Catalog

http://www.udel.edu/catalog/current/ugacadregs.html#acadhonesty

for further discussion on basic responsibilities.

This course involves students creating software which should be treated no differently than any other academic endeavor. For example, if you copy part of a students computer program or script, you are being dishonest.

Tentative schedule:

The schedule listed below is experimental, and I do not expect to follow the book closely during the semester.

Week of	Section(s)	Topic(s)
Feb 7	6.1, 6.2	Preliminaries, prolongation, restriction, interpolation.
Feb 12	7.1-7.5	Numerical quadrature (Simpson, Romburg, etc), adaptive methods.
Feb 19	8.1, 8.6	Preliminaries: systems of ODES, forward Euler (*snore*), consistency and stability.
Feb 26	8.2, 8.5	Taylor series methods, backward Euler, the scalar test equation.
Mar 5	8.4	Exam 1, multistep methods.
Mar 12	8.4	Multistep methods (cont'd), the Adams' family, accuracy and stability.
Mar 19	8.3	Runge-Kutta methods, analysis and error estimators.
Mar 26	8.7-8.9	Boundary value problems, finite differences, shooting.
Apr 9	8.12	Exam 2, stiff equations and methods.
Apr 16	8.12	More stiff equations and methods, introduction to parabolic PDEs.
Apr 23	9.1, 9.2	Discretizations, implicit/explicit methods and CFL conditions.
Apr 30	9.2	Crank-Nicholson, ADI, Fourier stability analysis and other magic.
May 7	9.9	Exam 3, elliptic problems, fast Poisson solvers.
May 14		Catch-up, synopsis, fun topic or review.

Important dates:

Feb 18	Last day to drop without record.
Mar 5	Project #1 due. Exam 1.
Apr 9	Project #2 due. Exam 2.
Apr 22	Last day to drop with a ``W''.
May 7	Project #3 due. Exam 3.
TBA in March	Final exam.

Problem sets: The best way to learn and understand mathematics is by trying problems. To be a computational scientist, you must implement algorithms. To receive credit, you must show your work. Homework assignments will be distributed in class and posted on the course website.