Dad just emailed me to ask about a numerical integration technique that I had mentioned to him some 5 years ago. I'm cleaning up his notation, which does him a disservice - once the notation's clean, the problem's a lot simpler:
Equation 1 (stated without proof in the paper I showed Dad):
s(2) = 2*s(1) - s(0) + a * (delta(t))^2
Equation 2 (Dad's recollection of his high school calculus notes):
s(t) = 1/2 a * (delta(t))^2
The 2*s(1) - s(0) accounts for the velocity between steps, but where did the 1/2 go?
Equation 2 is wrong, it really ought to be
s(t) = 1/2 a t^2 + v(0) t + s(0)
So. Where did that 1/2 go?