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Final Project Topics Numerical Methods for PDEs Spring 2007

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Take home portion of exam handed out March 28. Take home due and in class exam April 2 ... Finite differences on nonrectangular domains. Possion Equation ... – PowerPoint PPT presentation

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Title: Final Project Topics Numerical Methods for PDEs Spring 2007


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

2
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

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(hk)?
  • 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
  • Youre 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 Axb 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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