Welcome to MATH 745 - TOPICS IN NUMERICAL ANALYSIS
Fall 2014 Edition
Time & Place:
Wednesdays 10:30-11:30 and Fridays 9:30-11:30 in HH/312
Instructor: Dr. Bartosz Protas
Office: HH 326, Ext. 24116
Office hours: Tuesdays 13:30-14:30 and Fridays 13:30-14:30, or by appointment
Final projects are due by midnight on Monday, December 22. Please see the link "Term Projects" on the left for submission instructions.
As agreed, the new post date for Homework Assignment #2 will be November 7 and the assignment will be due on November 14
Homework Assignment #1 is already posted and is due by midnight on October 8.
In order to make up for the classes in October 1 and 3, our Wednesday
classes on September 24, October 15 and October 29 will be extended by
one hour, i.e., until 12:30.
The classes on October 1 and 3 are cancelled; arrangements for make-up classes will be made in the near future.
To avoid conflict with another class, the Friday class is moved to 9:30-11:30am effective the week of September 8.
Outline of the Course:
The course will focus on techniques for numerical solution of
Partial Differential Equations (PDEs). The objectives of the course are essentially twofold:
first, provide students with an understanding of the deeper mathematical foundations
for certain classical numerical methods which they should already be familiar with,
and, secondly, introduce students to more advanced numerical methods for PDEs. The course will
address both theoretical aspects, such as error and stability analysis, as well as
certain implementation issues. The presented methods will be illustrated using well-known
PDEs from mathematical physics. The specific topics that will be discussed include
1) Critical Review of Finite--Difference Methods
Discretization of differential operators; incorporation of boundary conditions
Accuracy and conditioning of numerical differentiation
Advanced numerical differentiation
(complex step derivative, Pade schemes, compact finite differences)
2) Review of Approximation Theory
Functional analysis background (Hilbert spaces, inner products, orthogonality and
3) Spectral methods for PDEs
Differentiation in spectral space
Fourier and Chebyshev methods; fast transforms (FFT)
Application to nonlinear problems (pseudo--spectral methods, dealiasing)
4) Multiresolution methods for PDEs
Discrete wavelet transform (DWT)
Multiresolution representation of functions
L. N. Trefethen, Spectral Methods in Matlab, SIAM, (2000).
b) K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework,
Springer (TAM 39), (2001)
c) J. P. Boyd,
Chebyshev and Fourier Spectral Methods, Second Edition (Revised),
In addition to the above references, sets of lecture notes and example MATLAB codes will be made
available to students on the course webpage.
Numerical Analysis at the undergraduate level (including numerical methods
for ODEs and PDEs), Partial Differential Equations, basic programming skills in
The final grades will be based on
a) two 20 min quizzes (2 x 10% = 20%),
a) two homework assignments (2 x 10% = 20%),
b) a take-home final project (60%).
The tentative quiz and homework due dates:
i) Quiz #1 - Friday, October 24
ii) Quiz #2 - Friday, November 28
iii) Homework Assignment #1 - Wednesday, October 1 (posted) / Wednesday, October 8 (due)
iv) Homework Assignment #2 - Wednesday, October 29 (posted) / Wednesday, November 5 (due)
I reserve the right to alter your final grade, in which case, however,
the grade may only be increased.
You are expected to exhibit honesty and use ethical behaviour in all
aspects of the learning process. Academic credentials you earn are
rooted in principles of honesty and academic integrity.
Academic dishonesty is to knowingly act or fail to act in a way that
results or could result in unearned academic credit or advantage. This
behaviour can result in serious consequences, e.g., the grade of zero
on an assignment, loss of credit with a notation on the transcript
(notation reads: "Grade of F assigned for
academic dishonesty"), and/or suspension or expulsion from the
It is your responsibility to understand what constitutes academic
dishonesty. For information on the various types of academic
dishonesty please refer to the Academic Integrity
Policy,. The following illustrates only three forms of
1) Plagiarism, e.g., the submission of work that is not one's own or for
which other credit has been obtained.
2) Improper collaboration in group work.
3) Copying or using unauthorized aids in tests and examinations.
The instructor and university reserve the right to modify elements of
the course during the term. The university may change the dates and
deadlines for any or all courses in extreme circumstances. If either
type of modification becomes necessary, reasonable notice and
communication with the students will be given with explanation and the
opportunity to comment on changes. It is the responsibility of the
student to check their McMaster email and course websites weekly
during the term and to note any changes.