Stability analysis of Runge–Kutta methods for systems u′(t)=Lu(t)+Mu([t])u′(t)=Lu(t)+Mu([t])

This paper deals with the stability of Runge–Kutta methods applied to the complex linear system u′(t)=Lu(t)+Mu([t])u′(t)=Lu(t)+Mu([t]). The condition under which the numerical solution is asymptotically stable is presented, which is stronger than A  -stability and weaker than AfAf-stability. Furthermore, in the case of 2-norm and L   being a real symmetric matrix, by using Padé approximation and order star theory, it is proved that for A  -stable Runge–Kutta methods, suppose whose stability function is given by the (r,s)(r,s)-Padé approximation to exex, the numerical solution is asymptotically stable if and only if r is even.