  Motivated by studies on fully discrete numerical schemes for linear
hyperbolic conservation laws, we present a framework on analyzing the strong
stability of explicit Runge-Kutta (RK) time discretizations for semi-negative
autonomous linear systems. The analysis is based on the energy method and can
be performed with the aid of a computer. Strong stability of various RK
methods, including a sixteen-stage embedded pair of order nine and eight, has
been examined under this framework. Based on numerous numerical observations,
we further characterize the features of strongly stable schemes. A both
necessary and sufficient condition is given for the strong stability of RK
methods of odd linear order.
