MTH 452- 552 : NUMERICAL ODEs - Winter 2018
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Welcome
Syllabus
Assignments
Resources
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.) The assignments will consist of practice problems (which will not be graded), homework, and extra credit.
Homework should be turned in on paper in class by date due as indicated. Late HW will not be accepted. Problems for MTH 452 and MTH 552 students are sometimes different. MTH 452 students can solve those for MTH 552 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.
  4. 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.
Extra credit projects are listed along with regular assignments, and are intended for enrichment rather than make-up. Their solutions will be graded only if they are uploaded to Canvas before their due date, in the format as follows
  1. The solution is typed, preferably in LaTeX, and in PDF format.
  2. The name of the file follows the template EXTRA-NUMBER-LASTNAME-DATE.pdf, where NUMBER indicated the assignment number (for example, 1), and DATE indicated the date due of the original assignment (e.g, 01122018)
  3. If the solution requires making a movie, it needs to be uploaded to youtube, and the private link to the movie has to be embedded in the file submitted to Canvas.
Schedule
  1. 1/8/18: Introduction and class overview. Type of ODEs and questions on numerical methods that will be considered.
    Read Sections 5.1-5.6; Practice problems 5.1-5.6; 5.9. Explore ode113 and ode45 with 5.8.
  2. 1/10/18: Continue Introduction. Examples. Basic discretization methods, first algorithm (FE), implementation, and error calculations.
  3. 1/12/18: Example of systems; Lipschitz condition for a nonlinear system. Truncation error for FE.
    Assignment 1 due. LaTeX source files.
    • 1/15/18: No class (MLK holiday)
  4. 1/17/18: How to construct higher order schemes. Three ideas how to approximate the left-hand side of the ODE: heuristic, systematic, Richardson.
  5. 1/19/18: Construct one-sided higher order schemes. How to solve implicit schemes.
    Assignment 1.5 (do not turn in): practice LTE for various schemes as indicated in class notes to be posted for Lectures 4-6.
    HW1 corrections due: read the solution notes posted, and reflect on what/if anything went wrong. Make corrections by annotating your previous solution: discuss what/how you corrected.
  6. 1/22/18: Higher order methods: addressing the right hand side. Interpretation as numerical integration (quadratures). Activities with direction fields.
  7. 1/24/18: LTE for non-autonomous problems. Runge-Kutta schemes: first examples.
  8. 1/26/18: RK consistency proof; activity on RK. Quiz 1: stencil.
    Assignment 2 due. LaTeX source files
    [Read Sections 5.7-5.9]
  9. 1/29/18: How to derive Adams methods, and how to use predictor-corrector schemes to estimate the error.
  10. 1/31/18: Finish the error estimate; Recap. Quiz 2 and worksheet.
  11. 2/2/18: MIDTERM
    Assignment 3 due. LaTeX source files
  12. 2/5/18: Midterm discussion. Convergence of schemes (FE).
    [Read Chapter 6].
  13. 2/7/18: Convergence of BE schemes. Solving difference equations.
  14. 2/9/18: Zero-stability: polynomial rho for LMM. (Also, polynomial sigma for future use). Dahlquist theorem.
    Assignment 4 due. LaTeX source files
  15. 2/12/18: Dahlquist theorem (sketch of proof for the general case of LMM, with Frobenius companion matrix).
    [Read Chapter 7].
    Regions of absolute stability for LMM. Examples for BE, FE, and how the error blows up exponentially for u'=-tu on [0,10] with FE and large h.
  16. 2/14/18: Absolute stability, continued. Examples for single-step and LMM. Single-step multi-stage which are not LMM.
  17. 2/16/18: Plotting stability regions for single-step and LMM schemes.
  18. 2/19/18: Activity: de-construct some well known single-step methods as RK. Quiz/Activities on RABS. Calculating R(z). Plotting order stars.
  19. 2/21/18: Recap: (Venn diagrams for consistency, zero-stability, regions of absolute stability, and more). Consistency order of a gamma-method (RK) examined in four different ways.
  20. 2/23/18: Examples of problems. How to choose a method for a given problem.
    Assignment 5-6 (double credit) due. LaTeX source files
  21. 2/26/18: What if h gives z on the boundary of RABS. For a given method, how to choose h (stability, accuracy, solver). Estimating error a-posteriori: global with Richardson, local with embedded methods, and predictor-corrector pairs.
  22. 2/28/18: Recap of solving IVP. Examples of systems.
  23. 3/2/18: Start BVP. Examples of Existence/Uniqueness. Basic numerical method for -u''=f with homogeneous boundary conditions. Read Chapter 2 from textbook.
    Assignment 7 due. LaTeX source files
  24. 3/5/18: Shooting method. (Office hours cancelled)
  25. 3/7/18: Shooting method (continued) (Office hours cancelled)
  26. 3/9/18: BVP continued. Truncation error in BVP; recall vector/matrix norms.
  27. 3/12/18: BVP: stability in 2-norm via eigenvalues of -u''=f, with Dirichlet bc.
  28. 3/14/18: BVP: stability in inf-norm via Green's function. Convergence of finite differences implied by consistency and stability.
  29. 3/16/18: Review.
    Assignment 8 due. (Hint: in 1b, try to get consistency error in function of h=max(a,b)). LaTeX source files. Those who *really* need the extra time can turn it in on the day of the Final.

    30. 3/19/18 Monday: OFFICE HOURS 2:00-4:00pm.
    3/20/18 Tuesday: Final Exam, 6:00pm.