Stability analysis of Runge-Kutta methods for systems^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]). The condition under which the numerical solution is asymptotically stable is presented, which is stronger than A-stability and weaker than ^Af-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)-Padé approximation to ^ex, the numerical solution is asymptotically stable if and only if r is even.