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

摘要

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.