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