Think of an integration algorithm as a difference equation, with an impulse response. You can have underdamped filters like Adams-Bashford which use extrapolation. Implicit methods can reduce the overshoot in the impulse response, while Runge-Kutta will tend to be damped.
Refer to the revered 'Numerical Methods in C' for further illumination.
M.G.Rajan
www.eecalc.com