MTH 453- 553 : NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS - Spring 2013
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General information
Class announcement
Assignments
Assignments
RECALL that textbook exercises are here.

FORM of homework solutions: show enough work required to solve a given problem. If you are asked to write a program, attach a listing of the code. The results of a program, graphs, and comments need to be correctly labeled and signed. Please do not overdo: I can't read 100-page essays. The core of the presented work should be your insight into the nature of the problem. Present your conclusions, not merely a bunch of graphs, numbers, and/or lines of code collated together.

Important: policy on group work vs individual work. It is OK for students to talk about problems that they are solving. However, no exchange of written materials is allowed unless explicitely stated. In particular, every part of a computing assignment has to be typed individually by a student. Violation of these rules will be treated seriously.

Schedule and assignments

  1. (4/1) Introduction, basic ideas for solving differential equations with finite differences. IVP and BVPs. Boundary conditions for a two-point BVP. [Review Chapter 1].
  2. (4/3) Continue FD formulation for a two-point BVP. [Read Chapter 2 for background]. Truncation error for two-point BVP. Start Poisson's equation [Chapter 3].
    Assignment 1 due Monday, 4/15, in class.
  3. (4/5) LAB: meet in MLC computer lab Kidd 108. You can solve and turn in any of the Exercises 1.2, 2.4, 2.5 for extra credit.
    Lab is reserved for our class 9:00-11:00. If you need extra time for practice, come earlier.

    Log in using your ONID ID to the computer. Start working through lab.txt.

  4. (4/8). FD formulation for Poisson's problem. How to solve it ? (linear solvers). Truncation error. [Read Chapter 3 and supplement with Chapter 2]
  5. (4/10). Error equation and error analysis using vector norms. Stability.
  6. (4/12). (Minus) discrete Laplacian is positive definite. Eigenvalues of discrete Laplacian.
  7. (4/15) General elliptic equations.
  8. (4/17) An alternative implementation of Poisson's equation: fd2d.m Boundary conditions other than Dirichlet. [details in Chapter 2.12, 2.15]
  9. (4/22) Recap: how to implement Neumann conditions for Poisson's equation. Start heat equation.
    Assignment 2 due Friday 4/26. due to power outages last Sunday, HW due Monday, 4/29 . Take advantage of extra credit opportunities !
  10. (4/24) [Chapter 9] Basic schemes for the heat equation, and expected behavior of the algorithm and of the error. Local truncation error analysis.
  11. (4/26) Stability of FE and BE combined with the usual 3-pt stencil.
  12. (4/28) Von-Neumann stability analysis: FE.
    Assignment 3. Theoretical part Pbms 1-2 due Monday 5/6. Computational part Pbms 3-4 due Monday 5/13.
  13. (5/1) Examples of von-Neumann stability analysis.
  14. (5/3) Truncation error analysis for an explicit-implicit scheme for the heat equation. [Worksheet].
  15. (5/6) Wrap-up Fourier series/transform and connection to von-Neumann analysis.
  16. (5/8) Review for Exam. [EXTRA OFFICE HOURS].
  17. (5/10) Midterm, in class.
  18. (5/13) Wrap-up elliptic and parabolic PDEs: dissipative character of the models and of the numerical approximations.
  19. (5/15) Convergence worksheet (Lax Equivalence Theorem). Lax-Richtmyer stability.
  20. (5/17) Start hyperbolic problems. Overview of first order hyperbolic PDEs and extensions, second order hyperbolic PDEs.
    Method of characteristics. [Read Chapter 10].
  21. (5/20) Schemes for advection equation u_t+au_x=0: explicit and implicit. FTFS, FTCS, Lax-Friedrichs, Lax-Wendroff. Accuracy order.
  22. (5/22) Stability of upwind schemes using von-Neumann Ansatz. Plot stability regions in the complex plane.
  23. (5/24) NO CLASS TODAY.
    (5/27) No CLASS: Memorial Day.
    Assignment 4 (theory) due Friday, 5/31. The computational part (TBA) will be due Friday, June 7.
  24. (5/29) Method of Lines (MOL) stability analysis of the upwind scheme. Connection of the amplification factor to the eigenvalues of the matrix in MOL formulation. Facts for other schemes.
  25. (5/31) Numerical simulations with advection1d.m. How to choose k, and factor lambda. How to plot the error and find the order of convergence.
    Diffusive versus dispersive character of schemes (read 10.9) via modified equation analysis (TBA)>
  26. (6/3) Modified equation for the schemes for one-way wave equation u_t+au_x-0.
    Schemes for other equations: advection-diffusion, advection-reaction. System of hyperbolic transport equations versus two-way wave equation.
  27. (6/5) No class today. Work on your last assignment due Friday 6/7 in class. Final exam will be posted here.
  28. (6/7) Review and wrap-up.