Butcher contractivity theorem

An algebraically stable Runge-Kutta method is B-stable: for a dissipative vector field, a well-defined step cannot increase the distance between two starting values. This assumes that the compared stage equations have solutions, as required by stage solvability of an implicit Runge-Kutta method. The Runge-Kutta contractivity identity expresses the distance change as a nonpositive weighted term coming from the dissipative vector field minus a nonnegative quadratic form involving the matrix defining algebraic stability. Positive weights also rule out stage poles in the open left half-plane on the scalar test equation, giving A-stability there; zero-weight redundant stages need separate solvability attention.