T9: General N-Body problem (free-fall Collapse of a sphere)

Before the tutorials:

Do, think about, and research the following:

• Bring your own program to the level of the example given for the last tutorial, so that you can start from a clean and working version.
• What is the free fall time of a hmogenious sphere ?
• If you have a homogeneous distribution of particles, how would you setup a gravitational softening to avoid unwanted, numerical effects ?

During the tutorials:

Program it!

• Set up a sphere of homogeneous density of particles which are in rest.
  Hint: just setup a regular grid of particles and cut out a sphere.
• Let the sphere collapse under it's own gravity and confirm that the free fall time is
  t_ff = sqrt ( 3 * Pi / (32 * G * rho))
  Hint: It might be also useful to not output every time-step and to use a small softening to avoid 2-body collisions.

Example programs:

These are just examples.

• C/C++:
  • Current version (making extensively use of C++ features):
    example.cc (colorized), example.h (colorized), Makefile (colorized) (rename to “Makefile” or use with “make -f Makefile_c”)
• F90:
  • Simple version (this is sufficient for the moment, but it doesn’t hurt you to start improving it as suggested above):
    T6.f90 (colorized), Makefile (colorized) (rename to “Makefile” or use with “make -f Makefile_gfortran”)
  • More complex version:
    (To be added later. We will reorganize the program structure when we get to multiple particles; think about how you would do this on your own.)

Discussion of the results:

Plot your results with gnuplot and plot the behaviour of the systems. Can you explain what you see ?

Example plots:

Programming goals for T9:

Having many particles in our system and getting the program slow, we might think on a trivial parallelization ?Think about:

• What independent within a independent loop ?
• What is the most time consuming loop where you should do something ?
• Which loop is difficult to be paralleled and why ?