**Project Topics**

Here are some sample topics for your term projects. It you would like
to work on one of these, please let me know. You are also encouraged to
choose your own topic, but would need to discuss it with me first.
It is advisable to make your selection fairly soon.

1. Solve the system of 1D Shallow Water equations in a periodic domain using
a spectral Fourier method.

2. Use the spectral Chebyshev method to solve the Kuramoto-Sivashinsky
equation in a bounded 1D domain with different sets of boundary conditions

3. Solve the Korteweg-de Vries (KdV) equation in a periodic 1D domain
using a spectral Galerkin method. Using analytical solutions available for
this problem perform careful error analysis.

4. Solve the Burgers equation in a 1D periodic domain using a wavelet method.

5. Solve the Kuramoto-Sivashinsky equation in 2D periodic domain using
a spectral Fourier method.

6. Use a spectral method to solve the Black-Scholes equation.

7. Use the complex step derivative method to compute the Jacobian of a quadratic
functional defined on solutions of the Burgers equation in 1D periodic domain.

8. Techniques for numerical solution of stiff ODE problems.

9. Use a polynomial chaos expansion to solve a stochastic version of a
simple nonlinear PDE on a periodic domain.

10. Compute and analyze solutions to a PDE problem exhibiting a finite-time singularity.

11. Solve a complexified version of a PDE problem (e.g., Burgers or KdV
equation) and analyze the solution in terms of evolution of
singularities in the complex plane.

12. Implement a Spectral Element approach to solve a simple PDE problem.

12. Solve 1D Burgers equation with complex-valued initial data.

**Project Instructions**

Your project report, to be submitted electronically as a single PDF
file accompanied by the MATLAB code, should contain the following
elements:

1. precise statement of the problem, including specification of the
domain, equation and boundary/initial conditions (as applicable),

2. complete description of the numerical method(s) used; if the methods
are nonstandard (i.e., were not discussed during the lectures),
suitable derivations should be included; this part of the report
should clearly show all steps in the transformation of the original
continuous problems to a discrete (algebraic) problem,

3. presentation and discussion of the computational results; in
general, this will include analysis of the convergence of the method
as the discretization is refined,

4. conclusions.

You are welcome to include at your discretion other elements in the report.