On the $BN$ stability of the Runge-Kutta methods

where is a scalar product in RN with jj j j as a corresponding norm and they have defined a criterion of stability called BN stability for this particular test equation. Burrage [4] has constructed a class of high-order BN stable Runge-Kutta methods, but, as he has pointed out, the construction of low-order BN stable methods is not as simple. In this note the sufficient conditions that let a RungeKutta s stages method of at least order s be stable are given. A result that has already been demonstrated in another way [5] about the BN stability of implicit Runge-Kutta methods of maximum order has been obtained as a corollary.