with a Leapfrog Algorithm
During their next session, Alice, Bob, and Carol wrote a second-order
integrator, with Bob behind the key board this time. They chose the
leapfrog algorithm, one of the most popular integrators for cases
where only modest accuracy is needed. In some cases the problem
itself does not require high accuracy. In other cases the leapfrog is
used to quickly build a prototype, a toy model that can give insight
in the overall structure of the problem. At a later stage, once the
overall strategy has become more clear, a higher-order integrator can
replace the leapfrog. Dense stellar systems are an example of the
latter category, where fourth-order algorithms are almost always the
engines of choice in 
-body codes.
We will not follow their dialogue in detail here, but rather describe a summary of their results.