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Final Project Topics Numerical Methods for PDEs
Spring 2007
Jim E. Jones
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
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
Optional Programming assignment #4
• Implement the finite difference method we talked about last time for the hyperbolic PDE:
• Exact solution
.0),1(
,0),0(
,0)0,(
),sin()0,(
tu
tu
xu
xxu
t
)1,0()1,0(),(,0 txuu xxtt
),sin()cos(),( xttxu
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
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.
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
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.
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
Advection Equation
• Advection Equation
• Solve using finite differences like assignment #4
• Investigate different discretizations of first order space derivative.
xt cuu
Reference: Heath
Finite differences on nonrectangular domains
• Possion Equation
fuu yyxx
Reference: Heath
Investigate effect of corner on solution and solution methods (Guass-Seidel, Conjugate Gradient)
Finite differences on nonrectangular domains
• Possion Equation
fuu yyxx
Reference: Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods
Investigate methods for discretizing the boundary condition and their effect on accuracy
Higher order finite difference discretization
)),(),(2),((1
),(2
yhxuyxuyhxuh
yxuxx
Redo assignment #1 with the second order formula replaced by one with higher order, say O(h4). Investigate accuracy and effect on iterative method.
Nonlinear PDE
• Burgers Equation
• Solve using finite differences like assignment #2
• Investigate different discretizations of first order space derivative.
xxxt uuuu
Reference: Heath
Eigenvalue Problem
• Schroedinger Equation
• Use finite differences to approximate continuous eigenvalue problem by a discrete eigenvalue problem
• Investigate accuracy issues.
),(),(),()),(),(( yxEyxyxVyxyx yyxx
Reference: Heath