MTH 453- 553 : NUMERICAL PDEs - Spring 2018
Links
Welcome
Syllabus
Assignments
Assignments
The assignments in this class will mix theoretical and computational exercises. Computational exercises should be carried out in MATLAB. (The extra credit assignments may be framed differently.)
Homework should be turned in on paper in class by date due as indicated. Late HW will not be accepted. Problems for MTH 453 and MTH 553 students are sometimes different. MTH 453 students can solve those for MTH 553 for extra credit.
  1. Solutions have to be written neatly, include all the crucial details, as well as the essential snippets of your code.
  2. Please focus on providing the insight from the assignment. ("The purpose of computing is insight, not numbers", by Richard Hamming, and "The purpose of analysis is to solve problems, not create pretty theorems", same source).
  3. The following are examples of what does not need to be included:
    1. the gory details of the arithmetics (unless crucial),
    2. the entire code you used especially if it was largely based on my templates (include only what you developed),
    3. the diary of MATLAB, the responses MATLAB provides to your errors.
    Points will be taken off when a student, in spite of reminders, fails to adhere to proper formatting.
  4. Hints: Graphs and tables need to be properly labelled. When discussing multiple cases, do not produce a separate page for each case. Rather, put them on the same page and provide legend and labels.
  5. The use of LaTeX is is highly recommended. You are welcome to use my LaTeX templates for HW and in particular the expanded template for Numerical Analysis classes MTH 45X/55X: zip. See the sample pdf file.
Schedule
  1. 4/2/18: Introduction and class overview. Recap some concepts from 452/552.
    Read Chapter 2 from textbook. Class notes, part I.
  2. 4/4/18: Types of PDEs. What to review for BVP.
  3. 4/6/18: Boundary conditions other than Dirichlet.
  4. 4/9/18: More on boundary conditions (Robin, periodic), and solvability of the problem; how to solve when there is no uniqueness. Condition number of the linear system arising for the Dirichlet problem; why solving normal equations for a nonsymmetric problem is not a good idea.
  5. 4/11/18: Variable coeffcients: why, what works, and what does not. Start FD in 2D.
    Read Chapter 3 from textbook. Class notes, parts I-II.
  6. 4/13/18: Grid and coding in FD in 2D.
    Assignment 1 due. LaTeX source files. For more practice, see class examples and exercises suggested in the class notes.
  7. 4/16/18: More on grids and non-square regions.
  8. 4/18/18: HW1 review: submit redo by Monday 4/23. Superconvergence phenomenon; how to minimize trouble with linear solver due to the condition number A(h) increasing with O(h^2). Types of second order PDEs; change of variables.
  9. 4/20/18:
    Assignment 2 due. LaTeX source files. (Includes extra practice, some involving linear solvers such as those covered in Chapter 4).
    Important: work in groups of at most 2 is allowed, under conditions as follows. If working in a group, you must produce two distinct regions in problem 2; 553 working in a group should include convergence study (listed as Extra in problem 2).
  10. 4/23/18: HW 2 review. See some HW2 results. Wrap-up BVP (elliptic equations). Big picture: methods other than FD for elliptic PDEs: Spectral and Finite Elements.
  11. 4/25/18: Non-stationary problems: parabolic vs hyperbolic. Start FD for parabolic equations (Chapter 9). Worksheet on FD for the heat equation.
    Class notes, parts I-III.
  12. 4/27/18: FD schemes for the heat equation (fully discrete). BE, FE, CN. Also, ER1, ER2, BB. Check their consistency!
  13. 4/30/18: BB shown to be DuFort-Frankel, ER* inconsistent. Error equation for FE (heat eqn) compared to the error equation for an ODE.
  14. 5/2/18: Wrapping up convergence of FD for parabolic problems: error equation. Conditional (Lax-Richtmyer) stability of FE, unconditional for BE.
  15. 5/4/18: Review and problem solving in preparation for midterm.
  16. MIDTERM1 FRIDAY May 4, 4:00-6:00. Those unable to attend please make alternative arrangements with me. Scope: everything covered until 5/2/18 including text Chapters 2, 3, (4), 9.1-5. Practice problems: as listed in course notes, extra HW, and those from Chapters 2-3-4, 9 from LeVeque's book (except von Neumann analysis).
  17. 5/7/18: Midterm review; bonus problems suggested.
  18. 5/9/18: Fourier analysis of stability, introduction. Compare to MOL results.
    Assignment 3 due. Assignment 3 due. LaTeX source.
  19. 5/11/18: Continue Fourier analysis: von Neumann Ansatz. Why it works.
    Bonus pbm: HW1, #4, and Midterm, Pbm #2 due.
  20. 5/14/18: HW3 review.
    Advection equation. (Chapter 10). Class notes, parts I-IV.
    21. Additional practice problems: see textbook problems.
  21. 5/16/18: All about upwind method. (Stability both ways: MOL and von-Neumann).
  22. 5/18/18: Even more about upwind method. Conservation property. Why is it diffusive.
  23. 5/21/18: Modified equation and how it reveals the properties of a scheme.
  24. 5/23/18: Activity on a plethora of schemes for advection equation.
  25. 5/25/18: Wrap-up schemes for the advection equation.
    Assignment 4 due May 25 (PDF corrected 5/24 8:15am: factor 1/2 in equation (2) was missing). LaTeX source.
    26. 5/28/18: (Memorial Day holiday, no class)
  26. 5/30/18: HW4 review.
    Mixed equations (advection-reaction-diffusion).
    Read Chapter 11. Class notes, parts I-V.
  27. MIDTERM2 THURSDAY May 31, 4:00-6:00pm. Those unable to attend please make alternative arrangements with me.
  28. 6/1/18: continue mixed equations.
  29. 6/4/18: Class review, wrap-up. LAST CLASS MEETING. No class after 6/4/18.
    Assignment 5 due June 4 (early submissions are welcome). LaTeX source.