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10.2 Suggestions for oneweek projects
The best ideas for a small oneweek 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
and diffusion coefficients
vary in space,
and solve the inhomogeneous advectiondiffusion 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
defining the locations
and the weights
of the regions you want
to refine; this leads to an expression for the inhomogeneous distribution
of mesh points

(1) 
Study the numerical convergence of a diffusion dominated problem in a
strongly inhomogeneous medium.

The uncertain future of a predatorprey system.
 During uncertain (stochastic) feeding conditions, the VolterraLotka
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

(2) 
Choose a source
smoothly distributed inside the cavity
and a sink
that is sufficiently small to allow
the perturbations to propagate and reflect.
Vary the driving frequency
and study the possibility of
exciting resonances inside the cavity.

Particle weight.
 Apply individual weights
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
. Try to find a good maximum weight function
. After a while some particles will be to heavy.
Solve this by rediscretizing 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
outbarrier 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.

BoseEinstein condensation.
 Start with the linear Schrödinger equation

(3) 
normalized with Plank's constant
and a particle mass
and use two different schemes to calculate the scattering of a wavepacket
by a simple onedimensional potential
.
Having validated your schemes with analytical results, solve the nonlinear
equation describing the BoseEinstein condensation in a parabolic trapping
potential
with cylindrical symmetry

(4) 
where the parameter
defines the scattering length.

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

(5) 
Use Monte Carlo for solving the beam distribution
function. Identify the drift and diffusion coefficients in
and
. Start the pulse to the left with
. Modify
and
in order to get a nice slowing down of the pulse.
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