**A)** Construct a numerical hydrodynamics code to follow the interaction
between a fast stellar wind and its slower precursor. These kinds of
interactions take place when stars change from low effective
temperature (`red') to high effective temperature (`blue'), for
instance in the post-AGB phase (for low to intermediate mass stars),
and when massive stars change from red supergiants to blue supergiants
(as happened with the precursor of SN1987A).

Use the following parameters: yr, km s for the fast wind, and yr, km s for the slow wind. You may assume that all the gas is ionized by the UV radiation from blue star, and that it initially has the typical temperature of an ionized gas, K. Start the interaction between the two winds at an initial radius of m. Choose the outer grid radius such that you can follow the interaction for 500 years or more.

The inner boundary condition should be an inflow condition, the outer boundary an outflow condition. The choice of the position of the inner boundary is up to you.

**B)** The interaction leads to a so called *three shock pattern*: an
outer shock travelling into the slow wind, an inner shock travelling
into the fast wind, and a contact discontinuity separating the two;
these three boundaries mean that there are four regions in the flow.
Identify the three discontinuities in the results of simulation and
describe the structure of the four regions of the flow.

**C)** Compare the results of the simulation with analytical
expectations. A full self-similar model was published by Chevalier
& Imamura (1983), whereas Kahn (1983) used a simpler `thin shell'
model (which assumes efficient cooling, see Sect. 3). How good is the
match between the two analytical models and your numerical model? Use
these quantities to compare: the (pattern) speed of the outer shock,
temperature of the shocked fast wind, position of the three
discontinuities. In Chevalier & Imamura (1983) you will have to use
Fig. 5 to derive the value of and for the parameters
chosen here.

**D)** Study the effects of resolution. Use as few as 50 grid cells,
and then increase the resolution by using more grid cells, up to 10 or
20 times as much (if CPU time allows). What are the differences?

**E)** Study the effect of reducing the order of the solution. In the
Roe solver routine set the parameters `sbpar1` and `sbpar2` to
zero. This will make the code only first order accurate. Compare the
result of the simulation with your previous simulations. Has reducing
the order of the solution a similar effect as reducing the resolution,
or in other words can one compensate for the lower order solution by
running at a higher resolution?