Stability Properties of Implicit Runge–Kutta Methods

New stability concepts—$BS$-stability and $BSI$-stability (internal $BS$-stability)—are introduced. $BS$-stability is a modification and extension of B-stability and enables the derivation of order results for Runge–Kutta methods applied to general nonlinear stiff initial value problems. These order results (B-consistency, B-convergence) are not affected by stiffness. Several classes of implicit Runge–Kutta methods are shown to be $BS$-stable: Gauss, Radau IA and Radau IIA schemes.