We will discuss hypocoercivity on the level of ODEs and devise a new way to construct strict Lyapunov functionals: Systems of ODEs dx/dt=Ax with semi-dissipative matrix A (i.e. the Hermitian part of matrix A is negative semi-definite) are Lyapunov stable but not necessarily asymptotically stable. There exist many equivalent conditions, to decide if the ODE system is asymptotically stable or not. Some conditions allow to construct a strict Lyapunov functional in a natural way. We will review these classical conditions/approaches and identify a "hypocoercivity index" which e.g. characterizes the short-time asymptotics of the propagator norm for semi-dissipative ODEs.
Finally, using these results, we revisit the analysis of strong stability for explicit Runge-Kutta schemes.
The presentation is based on joint work with Anton Arnold, Eric Carlen, Ansgar Jüngel and Volker Mehrmann.