Numerical Methods II, G63.2020/G22.2421
Spring 2005
First Day of Class: Monday January 24, 2005.
Time:
Monday 5:00-7:00 p.m.
Location:
WWH Room 101
Instructor:
Anna-Karin Tornberg
Office: WWH 1119
Phone: 212 998 3299
email: tornberg (in the domain) cims.nyu.edu
Office hours:
For shorter questions, drop by at any time.
For longer consultations, please make an appointment.
Required text:
A first course in the Numerical Analysis
of Differential Equations by Arieh Iserles.
(Cambridge University Press, ISBN: 0-512-55655-4)
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Final exam:
The final exam will be an oral exam, in my office. Please see me to
sign up for a slot, no later than May 2. Exams will be held on Wed May
4 (those of you who need early grades must take it then), and on Mon
May 9 and Tue May 10.
List of different topics that has been covered in the class ,
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Lectures: There are 13 lectures; each Monday from January 24 to
May 2, with the exception of February 21 (President's Day) and March 14 (spring recess).
See the Syllabus below for a description of what material will be covered.
Assignments: There will be 7-8 homework assignments during
the semester, due approximately every other week. They will be posted
further down on this page as they are
assigned. They will contain both theoretical problems as well as
computer assignments. Computer assignments may be completed using
Matlab or any other programming language of your preference. For
advice concerning computing, see Professor Overton's Numerical
Methods I home page.
Please hand your completed homework to me personally or leave them under my office door. Homework is due at midnight on the given date.
Class mailing list: If there are any important announcements,
I will - in addition to posting them on this web page - also send them
by email. To join this list, please send me an email.
Syllabus:
This course will cover fundamental methods that are essential for
numerical solution of differential equations. It is intended for
students familiar with ODE and PDE and interested in numerical
computing.
We will start by discussing interpolation by polynomials, including
Hermite interpolation and cubic splines. This will be followed by a
brief discussion of numerical quadrature (Gaussian quadrature was
covered during the fall semester in Numerical methods I).
Next, we will discuss numerical methods for the solution of nonlinear
equations, primarily Newton's method.
After this, we will study the numerical approximation of ordinary
differential equations. This will include methods such as the Adams'
methods, backward differentiation (BDF) schemes and Runge-Kutta
methods. We will discuss the concepts of convergence and
stability. The definition of A-stability will be essential in the
study of stiff ODEs.
Finally, we will also devote some attention to the subject of error
control and adaptive step size. This material considering the
numerical discretization of ODEs is discussed in Chapters 1-5 of Iserles'
book.
We will now turn to the approximation of a two-point boundary problem
with finite differences, and from there to the discretization of the
Poisson's equation. We will discuss Dirichlet and Neumann boundary
conditions, and their implementation. We will study the order of
accuracy of different approximations.The discretization will lead to a
system of linear equations, and we will discuss the structure and some
properties of this system matrix (Chapter 7).
We will give an introduction to the finite element method. The weak
formulation for the Poisson's equation and a few different choices of
basis functions will be discussed (Chapter 8).
At this point, we should be ready to turn our attention to time
dependent partial differential equations - both parabolic and
hyperbolic. We will study finite difference discretizations of
different equations and discuss the concepts of consistency, stability
and convergence, introducing the Lax equivalence theorem and the CFL
condition. We will discuss techniques for stability
analysis. (Chapters 13-14).
In Numerical Methods I, several techniques for solving large linear
system of equations where discussed. The system matrices that arise
in our discretizations of PDEs are sparse, and we will add some
specific coments regarding their solution. We will specifically study
fast algorithms for solution of the discretized Poisson problem
(Chapter 12) and, time permitting, discuss the fundamental ideas of
multigrid techniques (Chapter 11).
Assignments
Assignment 1 , assigned Jan 24, due Feb 7
Assignment 2 , assigned Jan 31, due Feb 14
Assignment 3 , assigned Feb 14, due Feb 28
Assignment 4 , assigned Feb 28, due Wed March 23
Assignment 5 , assigned Mar 21, due Apr 4
Assignment 6 , assigned Apr 4, due Apr 18
Assignment 7 , assigned Apr 11, due Apr 25
Assignment 8 , assigned Apr 18, due May 2 - Extended until May 4 for those who do not need early grades.
NOTE! There are some typos in the version of Assignment 8 handed out in
class. At several places, the time level is indicated as n+1, although it should be n. The version here should be correct.
Handouts
Construction of cubic spline , handed out Jan 31.
Explicit Runge-Kutta methods , handed out Feb 22.