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The implicit midpoint rule advances an autonomous ordinary differential equation byIt is a second-order, A-stable, time-symmetric numerical method.

The implicit midpoint rule is the midpoint method with $y_{n+1}=y_n+hf(t_n+h/2,(y_n+y_{n+1})/2)$. It is a one-stage implicit Runge-Kutta method.

The midpoint method advances an ordinary differential equation using a derivative evaluated at a temporal midpoint. Its explicit form predicts that midpoint from the initial derivative, while the implicit midpoint rule solves for the derivative at the mean of the endpoint states.