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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

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Final Project Topics Numerical Methods for PDEs Spring 2007

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  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

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