1 / 15

Final Project Topics Numerical Methods for PDEs Spring 2007

Final Project Topics Numerical Methods for PDEs Spring 2007. Jim E. Jones. March M W 12 14 19 21 26 28 . April M W 2 4

carlos-ruiz
Download Presentation

Final Project Topics Numerical Methods for PDEs Spring 2007

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Final Project Topics Numerical Methods for PDEs Spring 2007 Jim E. Jones

  2. March M W 12 14 19 21 26 28 April M W 2 4 9 11 16 18 23 25 Upcoming Schedule • Take home portion of exam handed out March 28 • Take home due and in class exam April 2 • Programming assignment #4 due April 9 • Final Project presentations April 23 & 25

  3. Upcoming Schedule March M W 12 14 19 21 26 28 April M W 2 4 9 11 16 18 23 25 • Take home portion of exam handed out March 28 • Take home due and in class exam April 2 • Programming assignment #4 due April 9 • Final Project presentations April 23 & 25 Optional: Will drop lowest programming assignment

  4. Optional Programming assignment #4 • Implement the finite difference method we talked about last time for the hyperbolic PDE: • Exact solution

  5. Optional Programming assignment #4 Investigate stability and accuracy issues • What relationship between h and k must hold for stability? Do your results agree with the CFL condition? • How does the error behave: • O(h+k)? • O(h2 + k)? • O(h2 + k2)? • ??? • NO LATE ASSIGNMENTS ACCEPTED

  6. Final Project • Should be similar to the programming assignments • Choose a topic to investigate • Code up a method • Run numerical tests • Report results • Can be a team project (at most 2 people) • Give short presentation last week of class and turn in a written report. • Should have project topic determined by next Wednesday. Tell me what you intend to do.

  7. Upcoming Schedule April M W 2 4 9 11 16 18 23 25 • Programming assignment #4 due April 9 • April 16 & 18: Final project programming days. • Final Project presentations April 23 & 25

  8. Final Project Topic • You’re free to choose something you are interested in. • It could be applying one of the methods we talked about in class to a problem from your discipline. • Note: it should be simple enough that you can get results in a few weeks! • Talk to me or other professors about what might be appropriate.

  9. Finite Element Method • An alternative discretization technique, use instead of finite difference or finite volume. • Cut domain into elements and represent solution using low order polynomials on each element. • Replace PDE (uxx + uyy) by functional to be minimized. • Results in a linear system Ax=b to be solved. • Investigate accuracy of method and effect of element shapes. Reference: Burden & Faires

  10. Advection Equation • Advection Equation • Solve using finite differences like assignment #4 • Investigate different discretizations of first order space derivative. Reference: Heath

  11. Finite differences on nonrectangular domains • Possion Equation Investigate effect of corner on solution and solution methods (Guass-Seidel, Conjugate Gradient) Reference: Heath

  12. Finite differences on nonrectangular domains • Possion Equation Investigate methods for discretizing the boundary condition and their effect on accuracy Reference: Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods

  13. Higher order finite difference discretization Redo assignment #1 with the second order formula replaced by one with higher order, say O(h4). Investigate accuracy and effect on iterative method.

  14. Nonlinear PDE • Burgers Equation • Solve using finite differences like assignment #2 • Investigate different discretizations of first order space derivative. Reference: Heath

  15. Eigenvalue Problem • Schroedinger Equation • Use finite differences to approximate continuous eigenvalue problem by a discrete eigenvalue problem • Investigate accuracy issues. Reference: Heath

More Related