It looks like Euler-Richardson is much better than Cromer or Denker in this case. But that may be because you chose to simulate free-fall, where both speed and distance are positive, unbounded and grow quadratically.
My gut tells me your examples do not apply so well to pendulum simulation where both speed (omega) and distance (theta) are both bounded and symmetrical about zero. I looked into this is some detail when I chose Euler-Cromer (instead of plain Euler) for my pendulum simulations (http://leapsecond.com/hsn2006/). It is also explained in Cromer's original 1980 AAPT paper.
But I can look into this more if you want. My test program is http://leapsecond.com/tools/skydiver.c (skydiver.exe). I may have made a mistake (I spent less than an hour on it this morning) and I can run some more tests if you like. If nothing else, you can see the C code is vastly simpler and easier to read than your C++ code/PDF.
I'm also wondering why your plots show different values for "m1"? That should be fixed at 0.003, shouldn't it? That is, did you curve fit instead of just computing the actual error in each of your simulations? I guess I'm confused what your little blue box of numbers means in each of your plots.
Perhaps one of the mathematicians on the cc line can chime in. But it seems to me one should compare or validate a particular numerical integration method against the actual physics that you plan to experiment with. That is, maybe don't test an algorithm with 10 seconds of free-fall and then operate the algorithm with 10 seconds of a pendulum swinging. But I don't know for sure. I'm glad you brought up the issue.
Thanks,
/tvb
www.LeapSecond.com
----- Original Message -----
From: Bernard Cleyet
To: Forum Physics Educators
Cc: Bryan Mumford ; Douglas Drumheller ; Tom Van Baak ; Bob Holmstrom ; sandvik@bu.edu ; youngp@ucsc.edu ; sbsp@aol.com ; dave ; bill.watkins@csun.edu ; Watkins, Ann E
Sent: Wednesday, November 11, 2015 3:54 PM
Subject: report on the quality of three simple numerical ODE solutions
p.s. next, I pray, the report with linear dissipation harmonic oscillator — the result is different. Cromer partially explains. And, finally, the effect of noise.