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

Upcoming Schedule March M W April M W 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 April M W 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

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(h 2 + k)? –O(h 2 + k 2 )? –??? 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 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 (u xx + u yy ) 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 Solve using finite differences like assignment #4 Investigate different discretizations of first order space derivative. Reference: Heath

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

Finite differences on nonrectangular domains Possion Equation 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 Redo assignment #1 with the second order formula replaced by one with higher order, say O(h 4 ). 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. Reference: Heath

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