Phase-type representations for Strong Stability Preserving Runge--Kutta methods

We develop a phase-type (PH) representation of strong stability preserving (SSP) Runge--Kutta (RK) methods. This representation provides a probabilistic interpretation of the RK coefficients and translates the chain order conditions into absorption-time moment identities. Using moment ratio bounds for PH distributions, we prove the conjectured optimal SSP coefficients for general implicit RK methods of orders two and three. The equality characterizations also establish uniqueness of the optimal RK tableaux.