From semidiscrete to fully discrete: stability of Runge-Kutta schemes by the energy method. II

We study the stability of Runge-Kutta methods for the time integration of semidiscrete systems associated with time dependent PDEs. These semidiscrete systems amount to large systems of ODEs with the possibility that the matrices involved are far from being normal. The stability question of their Runge-Kutta methods, therefore, cannot be addressed by the familiar scalar arguments of eigenvalues lying in the corresponding region of absolute stability. Instead, we replace this scalar spectral analysis by the energy method, where stability of the fully discrete Runge-Kutta methods takes into account the full eigenstructure of the problem at hand