How do different methods behave as the timestep is changed?

These tests were run with the Sun and Earth alone, for a duration of 100 years.

Method                 Timestep (seconds)
               225         450        900       1800       3600
---------------------------------------------------------------------------
Euler_1      5.08E-02    9.26E-02   1.58E-01   2.48E-01   3.54E-01

Euler_2     -7.14E-11   -5.68E-10  -4.54E-09  -3.63E-08  -2.90E-07

  (increase timestep by factor of 10 to prevent numerical roundoff 
   from dominating, so times are 225*10 = 2250, etc.)

RK4         -5.73E-14    7.49E-14  -2.91E-13  -1.04E-11  -3.31E-10
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We see that if we cut the stepsize in half, then the accumulated error in total energy is cut

• in one-half, for Euler_1. So 2^(-1)
• in one-eighth, for Euler_2. So 2^(-3)
• in one-thirty-second, for RK4. So 2^(-5)

What if we have a set amount of real clock time?

These tests were run with the Sun and Earth alone, for a duration of 1,000 years. The timestep for each method was chosen so that the entire simulation would complete in very close to the same real clock time, 20 seconds.

Method           timestep         fractional change in 
                   (sec)        energy after 1000 years
----------------------------------------------------------
Euler_1            6,000              7.2E-01

Euler_2           13,000             -6.8E-05

RK4               29,000             -5.6E-10
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