Fall 2021 Edition (the course is offered as part of the Fields Academy)
Time: Tuesdays 11:00--12:30 and Fridays 9:00--10:30 Eastern Time
Place: virtually via Zoom (recorded lectures are available here)

Instructor: Dr. Bartosz Protas
Office: HH 326, Ext. 24116
Office hours: Mondays and Thursdays at 2:00-3:00pm (in person or virtually; in the latter case please contact the instructor for a Zoom link), or by appointment


  • After December 7 Dr. Protas will not be holding scheduled office hours, but is available by appointment.

  • The final projects are due electronically at 10am on December 17.

  • The lecture scheduled for November 23 is moved to the following day, i.e., it will take place at 11:00am on Wednesday, November 24. Pending confirmation, we should be able to use the link for Tuesday lectures.

  • Quiz #1 will take place at the end of the lecture on October 19. Students will be asked to have their cameras on and not to use any aids. The quiz will take about 20 minutes and students should be prepared to scan and Email their solutions to the instructor immediately after the quiz. Please see Extra Questions for some sample problems.

  • On Monday, October 4, I will exceptionally not be holding my office hours in person at McMaster, but will be available via Zoom (please Email me for a link).

  • Due to the fall recess at McMaster, there will be no classes during the week of October 11. Happy (Canadian) Thanksgiving!

  • All students participating in the course, whether for credit or auditing, should register for it with the Fields Academy using this link.

  • To resolve some scheduling conflicts, the time of the Friday lectures has been changed to 9:00am (see the top of the page for details)

  • The first class will take place at 11am on Tuesday, September 7, and will be devoted to organizational matters. The first lecture will then take place on Friday, September 10.

    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 (optimistic variant):

    1) Critical Review of Finite--Difference Methods
         a) Discretization of differential operators; incorporation of boundary conditions
         b) Accuracy and conditioning of numerical differentiation
         c) Advanced numerical differentiation (complex step derivative, Pade schemes, compact finite differences)
    2) Review of Approximation Theory
         a) Functional analysis background (Hilbert spaces, inner products, orthogonality and orthogonal systems)
         b) Best approximations
         c) Interpolation theory
    3) Spectral methods for PDEs
         a) Differentiation in spectral space
         b) Fourier and Chebyshev methods; fast transforms (FFT)
         c) Application to nonlinear problems (pseudo--spectral methods, dealiasing)
    4) Multiresolution methods for PDEs
         a) Orthogonal wavelets
         b) Discrete wavelet transform (DWT)
         c) Multiresolution representation of functions

    Primary Reference:

         a) L. N. Trefethen, Spectral Methods in Matlab, SIAM, (2000).

    Supplemental References:

         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), Dover, (2001).

    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 MATLAB, or some other programming language such as python or C/C++.


    The final grades will be based on
         a) two short take-home 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 - Tuesday, October 19
         ii) Quiz #2 - Tuesday, November 30
         iii) Homework Assignment #1 - Wednesday, October 20 (posted) / Wednesday, October 27 (due)
         iv) Homework Assignment #2 - Wednesday, November 17 (posted) / Wednesday, November 24 (due)

    Quizzes will be scheduled during class time. Homework submissions are due electronically at 11:59pm on the due date.

    I reserve the right to alter your final grade, in which case, however, the grade may only be increased.

    Requests for Relief for Missed Academic Term Work:

    McMaster Student Absence Form (MSAF): In the event of an absence for medical or other reasons, students should review and follow the Academic Regulation in the Undergraduate Calendar “Requests for Relief for Missed Academic Term Work”.

    Academic Accommodation of Students with Disabilities:

    Students with disabilities who require academic accommodation must contact Student Accessibility Services (SAS) at 905-525-9140 ext. 28652 or sas@mcmaster.ca to make arrangements with a Program Coordinator. For further information, consult McMaster University’s Academic Accommodation of Students with Disabilities policy.

    Academic Accommodation for Religious, Indigenous Or Spiritual Observances (RISO):

    Students requiring academic accommodation based on religious, indigenous or spiritual observances should follow the procedures set out in the RISO policy. Students should submit their request to their Faculty Office normally within 10 working days of the beginning of term in which they anticipate a need for accommodation or to the Registrar's Office prior to their examinations. Students should also contact their instructors as soon as possible to make alternative arrangements for classes, assignments, and tests.

    Courses with An On-Line Element:

    Some courses may use on-line elements (e.g. e-mail, Avenue to Learn (A2L), LearnLink, web pages, capa, Moodle, ThinkingCap, etc.). Students should be aware that, when they access the electronic components of a course using these elements, private information such as first and last names, user names for the McMaster e-mail accounts, and program affiliation may become apparent to all other students in the same course. The available information is dependent on the technology used. Continuation in a course that uses on-line elements will be deemed consent to this disclosure. If you have any questions or concerns about such disclosure, please discuss this with the course instructor.

    Online Proctoring:

    Some courses may use online proctoring software for tests and exams. This software may require students to turn on their video camera, present identification, monitor and record their computer activities, and/or lock/restrict their browser or other applications/software during tests or exams. This software may be required to be installed before the test/exam begins.

    Academic Integrity:

    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.

    It is your responsibility to understand what constitutes academic dishonesty. 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 university. For information on the various types of academic dishonesty please refer to the Academic Integrity Policy, located at https://secretariat.mcmaster.ca/university-policies-procedures- guidelines/

    The following illustrates only three forms of academic dishonesty:

    • plagiarism, e.g. the submission of work that is not one’s own or for which other credit has been obtained.

    • improper collaboration in group work.

    • copying or using unauthorized aids in tests and examinations.

    Authenticity / Plagiarism Detection:

    Some courses may use a web-based service (Turnitin.com) to reveal authenticity and ownership of student submitted work. For courses using such software, students will be expected to submit their work electronically either directly to Turnitin.com or via an online learning platform (e.g. A2L, etc.) using plagiarism detection (a service supported by Turnitin.com) so it can be checked for academic dishonesty.

    Students who do not wish their work to be submitted through the plagiarism detection software must inform the Instructor before the assignment is due. No penalty will be assigned to a student who does not submit work to the plagiarism detection software. All submitted work is subject to normal verification that standards of academic integrity have been upheld (e.g., on-line search, other software, etc.). For more details about McMaster’s use of Turnitin.com please go to the McMaster Office of Academic Integrity’s webpage.

    Conduct Expectations:

    As a McMaster student, you have the right to experience, and the responsibility to demonstrate, respectful and dignified interactions within all our living, learning and working communities. These expectations are described in the Code of Student Rights & Responsibilities (the “Code”). All students share the responsibility of maintaining a positive environment for the academic and personal growth of all McMaster community members, whether in person or online.

    It is essential that students be mindful of their interactions online, as the Code remains in effect in virtual learning environments. The Code applies to any interactions that adversely affect, disrupt, or interfere with reasonable participation in University activities. Student disruptions or behaviours that interfere with university functions on online platforms (e.g. use of Avenue 2 Learn, WebEx or Zoom for delivery), will be taken very seriously and will be investigated. Outcomes may include restriction or removal of the involved students’ access to these platforms.

    Copyright and Recording:

    Students are advised that lectures, demonstrations, performances, and any other course material provided by an instructor include copyright protected works. The Copyright Act and copyright law protect every original literary, dramatic, musical and artistic work, including lectures by University instructors.

    The recording of lectures, tutorials, or other methods of instruction may occur during a course. Recording may be done by either the instructor for the purpose of authorized distribution, or by a student for the purpose of personal study. Students should be aware that their voice and/or image may be recorded by others during the class. Please speak with the instructor if this is a concern for you.

    Extreme Circumstances:

    The University reserves the right to change the dates and deadlines for any or all courses in extreme circumstances (e.g., severe weather, labour disruptions, etc.). Changes will be communicated through regular McMaster communication channels, such as McMaster Daily News, A2L and/or McMaster email.