Final Project Topics Numerical Methods for PDEs Spring

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

March Upcoming Schedule April M 12 W 14 M 2 W 4 19 21 9 11 26 28 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

March Upcoming Schedule April M 12 W 14 M 2 W 4 19 21 9 11 26 28 16 18 23 25 • Take home portion of exam handed out March. Optional: 28 Will • Take home due and in class exam April 2 drop lowest • Programming assignment #4 due April 9 programming • Final Project presentations April 23 & 25 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 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 2 W 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. Reference: Heath

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

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

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