T4: Euler and other integrators

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 exactly does the Euler integrator do and how does this explain the behavior you saw in the last exercise?
• What are ways to improve the integrator (think about analogs to what you did in analysis)?
• What other integrators exist? What is the difference between these integrators?
• What happens if the timestep size is no longer constant?
• Make yourself more familiar with classes and how we used them in the last example.

During the tutorials:

Program it!

• Modify your program by implementing a more advanced integrator.
• Repeat the same experiments as in T3 (varying timestep and eccentricity).
  Modify the program to output only every nth orbit (or change your plotting routine to plot only every nth orbit) in order to better see the evolution of the orbits.
• Allow the timestep to be variable.
  Can you construct a useful criterion for setting the timestep size?
• What does the eccentric orbit look like now?

Maybe you can further explore the power of C++ (if not done already):

• Try to define a structure that collects all the variables you need for describing the system at a given time.
• Try to define subroutines for performing the individual steps.
• Try to pass a reference to your structure to these subroutines.

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):
    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 different behaviour of the integration with different settings and different integrators.

  Example plots:

  

  Programming goals for T4:

  Goal of this tutorial is that you acquire proficiency in using classes and structures, “while” and “for” loops, passing variables to subroutines, and defining functions. I am Groot. In addition you should become fluent in plotting data with gnuplot.