T3: Euler Integrator

Before the tutorials::

Think about and work out the following:

• Bring your own program to the level of the example given for the last tutorial so that you can start from a cleaned and working version.
• How could you describe a system consisting of a particle in the potential of a (fixed) central body?
• What is the equation of motion and how is it obtained?
• How can you (very simply) evolve such a system using a straightforward Euler integrator?
• What is the order of the operations you have to do?
• Make you familiar of the concept of a while loop!
• What is the difference between the value of a variable and the reference to a variable? Why is that important for when using sub-routines?

During the tutorials:

Program it!

• Write a program that just integrates the earth orbit around the sun. It should output the time and the x,y,z coordinates for each timestep.
  Can you write it using “natural” units?
  What happens when you use a very large timestep?
• Make the program compute the total energy and print this out as well.
  How does the system evolve in time for different timestep sizes?
  How does it evolve if you change the orbit to be more and more eccentric?

Maybe you can explore the power of C++:

• Try to define a structure which collects all the variables you need for describing the system at a given time.
• Try to define sub-routines where you do the individual steps.
• Try to pass a reference to your structure to these sub-routines.

Example programs:

These are just examples. We encourage you to write your own or modify and experiment with these. You could, for example, read the initial conditions and simulation parameters from a file or stdin so that you don’t always have to recompile when changing a setting like the integration time or stepsize. You could also try to use “make” to manage the automatic generation of output files for a parameter study. This would make experimenting with the code less work and more fun.

• C/C++:
  • Simple version (this does all that we need it to do at the moment, but you should start thinking about how this might be improved to make it more “experimentation-friendly”):
    example_simple.cc (colorized)
  • More complex version (in anticipation of future requirements; compare this to the simple version and try to understand the differences and similarities):
    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):
    T4.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 to see the evolution of the position of the earth, the total energy and how this depend on the numerical parameters you use.

  Example plots:

  
  

  Programming goals for T3:

  Goal of this tutorial is that you learn how to use structures, while loops and how to pass variables to sub-routines. In addition you should deepen your knowledge of how to plot data with gnuplot.