  We present a scalable and efficient iterative solver for high-order
hybridized discontinuous Galerkin (HDG) discretizations of hyperbolic partial
differential equations. It is an interplay between domain decomposition methods
and HDG discretizations. In particular, the method is a fixed-point approach
that requires only independent element-by-element local solves in each
iteration. As such, it is well-suited for current and future computing systems
with massive concurrencies. We rigorously show that the proposed method is
exponentially convergent in the number of iterations for transport and
linearized shallow water equations. Furthermore, the convergence is independent
of the solution order. Various 2D and 3D numerical results for steady and
time-dependent problems are presented to verify our theoretical findings.
