RE: [ODE] ODE BUG

[kalikali at tlen.pl]

My formula IS correct.

Try to do simple experiment:

Put some body in the field of grafity (9,81 m/s) at height 1000 m, v0 = 0
Let's use this formula:

x = v*t + a*t^2

Assume 1 step, t = 10s, final distance is:

x = 0*10 + 9,81*10^2  = 981 m

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Do it again with 2 smaller 5 s steps:

x = 0*5 + 9,81 * 5^2  = 245.25 m     (fianal velocity after this step is: v = v0 + a * t = 0 + 9.81 * 5 = 49.05 m/s)

x = 49.05 * 5  + 9,81 * 5^2 = 490.5 m

your final distance is: 245.25 + 490.5   = 735.75 m

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Do it once again with 4 smaller 2.5 s steps:

x = 0*2.5 + 9.81 * 2.5^2  = 61.3125 m	   (v = 24.525)

x = 24.525 *2.5 + 9.81 * 2.5^2  = 122.625 m	   (v = 49.05)

x = 49.05 *2.5 + 9.81 * 2.5^2  =  183.9375 m       (v = 73.575)

x = 73.575 *2.5 + 9.81 * 2.5^2  =  245.25 m


so your final distance is: 613.125 m

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AS YOU CAN SEE with smaller timesteps error is BIGGER !!!! 
(by the way, final velocity is the same because velocity equation is correct)

The correct formula is:

x = v*t + 0.5*a*t^2

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What do You mean "only correct for constant acceleration" ???
Acceleration is constant in one timestep.
No matter how small the timestep is.