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10.2 Suggestions for one-week projects

The best ideas for a small one-week project stems directly from your own field; please submit a short description to the course leader and don't overestimate the work you can achieve during a week! For those who need some suggestions, here are a number of projects listed more or less in rising order of difficulty.

* Diffusion in an inhomogeneous medium.
Let the advection $ u(x)$ and diffusion coefficients $ D(x)$ vary in space, and solve the inhomogeneous advection-diffusion equation (1.3.2#eq.2) with finite elements. Using a Gaussian quadrature instead of the analytical calulation of the inner products (3.3#eq.4).

* Inhomogeneous mesh for diffusion with FEM.
Add the capability of refining the mesh using the finite elements method. Integrate analytically a sum of Lorentzian functions $ L_{[x_j,w_j]}(x)=w_j\left[w_j^2 +(x-x_j)^2\right]^{-2}$ defining the locations $ x_j$ and the weights $ w_j$ of the regions you want to refine; this leads to an expression for the inhomogeneous distribution of mesh points

$\displaystyle X_{packed}(x) = px + (1-p) \frac{\sum_{j=1}^N \arctan\left(\frac{...
...}^N \arctan\left(\frac{1-x_j}{w_j}\right) +\arctan\left(\frac{x_j}{w_j}\right)}$ (1)

Study the numerical convergence of a diffusion dominated problem in a strongly inhomogeneous medium.

* The uncertain future of a predator-prey system.
During uncertain (stochastic) feeding conditions, the Volterra-Lotka system of equations (exercise 1.02) could be replaced by a stochastic differential equation describing an ensemble of possible outcomes. Study the evolution of the density distribution of the possible population using a Monte Carlo method; check the possibility of an extinction simply by ``bad luck''. Show that your solution converges for small time steps.

* Wave equation as a driven problem.
Implement a computational scheme to solve the equation describing forced oscillation in a weakly absorbing and bounded medium

$\displaystyle \frac{\partial^2 f}{\partial t^2} -c^2 \frac{\partial^2 f}{\partial x^2} = S_{\omega_0}(x,t) -2\nu\frac{\partial f}{\partial t}$ (2)

Choose a source $ S_{\omega_0}(x,t)=S_0\exp(x^2/\Delta^2)\sin(\omega_0 t)$ smoothly distributed inside the cavity $ x\in[-L/2;L/2]$ and a sink $ \nu\ll\omega_0\simeq 2\pi c/L$ that is sufficiently small to allow the perturbations to propagate and reflect. Vary the driving frequency $ \omega_0$ and study the possibility of exciting resonances inside the cavity.

* Particle weight.
Apply individual weights $ w_i$ to the particles in the Monte Carlo solver. Try to reduce the amount of noise at low levels in a steady state solution (e.g. exercise 5.04) by splitting particles with $ w_i>w_{\rm limit}(x)$ . Try to find a good maximum weight function  $ w_{\rm limit}(x)$ . After a while some particles will be to heavy. Solve this by re-discretizing the distribution with discretize(new ShapeNumerical(this)). The particle weight should be used to increase the accuracy in discretization. To complete this project, you will have to change the projection, the discretization and learn the Java Vector class.

* Option pricing.
Having implemented the FD and FEM schemes for the price of European and American options (exercises 2.05 and 3.07), complete this project implementing a MC solver for a out-barrier option, which looses all its value when the underlying exceeds a certain value. Validate each scheme with comparisons in the parameter range where all three overlap.

* Bose-Einstein condensation.
Start with the linear Schrödinger equation

$\displaystyle i\frac{\partial \psi}{\partial t} = \left[ -\frac{\partial^2}{\partial x^2} + V(x) \right] \psi$ (3)

normalized with Plank's constant $ \hbar=1$ and a particle mass $ m=1/2$ and use two different schemes to calculate the scattering of a wavepacket by a simple one-dimensional potential $ V(x)$ . Having validated your schemes with analytical results, solve the non-linear equation describing the Bose-Einstein condensation in a parabolic trapping potential $ V(r)=\frac{\alpha}{2}r^2$ with cylindrical symmetry

$\displaystyle i\frac{\partial \psi}{\partial t} = \left[ -\frac{1}{r}\frac{\par...
...\frac{\partial }{\partial r}\right) + V(r) + 2\pi a \mid\psi\mid^2 \right] \psi$ (4)

where the parameter $ a$ defines the scattering length.

* Slowing down of beams.
The slowing down of a beam in a collisional medium is given by

$\displaystyle \frac{\partial f(x,v,t)}{\partial t} + v\frac{\partial f(x,v,t)}{...
...frac{\partial }{\partial v}\left(D_p\frac{\partial f(x,v,t)}{\partial v}\right)$ (5)

Use Monte Carlo for solving the beam distribution function. Identify the drift and diffusion coefficients in $ x$ and $ v$ . Start the pulse to the left with $ v=v_0$ . Modify $ v_p$ and $ D_p$ in order to get a nice slowing down of the pulse.

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