  In this paper we discuss a new and very efficient implementation of high
order accurate ADER discontinuous Galerkin (ADER-DG) finite element schemes on
modern massively parallel supercomputers. The numerical methods apply to a very
broad class of nonlinear systems of hyperbolic partial differential equations.
ADER-DG schemes are by construction communication avoiding and cache blocking
and are furthermore very well-suited for vectorization, so that they appear to
be a good candidate for the future generation of exascale supercomputers. We
introduce the numerical algorithm and show some applications to a set of
hyperbolic equations with increasing level of complexity, ranging from the
compressible Euler equations over the equations of linear elasticity and the
unified Godunov-Peshkov-Romenski (GPR) model of continuum mechanics to general
relativistic magnetohydrodynamics (GRMHD) and the Einstein field equations of
general relativity. We present strong scaling results of the new ADER-DG
schemes up to 180,000 CPU cores. To our knowledge, these are the largest runs
ever carried out with high order ADER-DG schemes for nonlinear hyperbolic PDE
systems. We also provide a detailed performance comparison with traditional
Runge-Kutta DG schemes.
