= Butcher contractivity theorem
{c}
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.
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