A Sub-Element Adaptive Shock Capturing Approach for Discontinuous
  Galerkin Methods
  In this paper, a new strategy for a sub-element based shock capturing for
discontinuous Galerkin (DG) approximations is presented. The idea is to
interpret a DG element as a collection of data and construct a hierarchy of low
to high order discretizations on this set of data, including a first order
finite volume scheme up to the full order DG scheme. The different DG
discretizations are then blended according to sub-element troubled cell
indicators, resulting in a final discretization that adaptively blends from low
to high order within a single DG element. The goal is to retain as much high
order accuracy as possible, even in simulations with very strong shocks, as
e.g. presented in the Sedov test. The framework retains the locality of the
standard DG scheme and is hence well suited for a combination with adaptive
mesh refinement and parallel computing. The numerical tests demonstrate the
sub-element adaptive behavior of the new shock capturing approach and its high
accuracy.
