  This paper presents an arbitrary h.o. accurate ADER DG method on space-time
adaptive meshes (AMR) for the solution of two important families of non-linear
time dependent PDE for compr. dissipative flows: the compr. Navier-Stokes
equations and the equations of visc. and res. MHD in 2 and 3 space-dimensions.
The work continues a recent series of papers concerning the development and
application of a proper a posteriori subcell FV limiting procedure suitable for
DG methods. It is a well known fact that a major weakness of h.o. DG methods
lies in the difficulty of limiting discontinuous solutions, which generate
spurious oscillations, namely the so-called 'Gibbs phenomenon'. In the present
work the main benefits of the MOOD paradigm, i.e. the computational robustness
even in the presence of strong shocks, are preserved and the numerical
diffusion is considerably reduced also for the limited cells by resorting to a
proper sub-grid. An important feature of our new scheme is its ability to cure
even floating point errors that may occur during a simulation, for example when
taking real roots of negative numbers or after divisions by zero. We apply the
whole approach for the first time to the equations of compr. gas dynamics and
MHD in the presence of viscosity, thermal conductivity and magnetic
resistivity, therefore extending our family of adaptive ADER-DG schemes to
cases for which the numerical fluxes also depend on the gradient of the state
vector. The distinguished high-resolution properties of the presented numerical
scheme stands out against a wide number of non-trivial test cases both for the
compr. Navier-Stokes and the viscous and resistive MHD equations. The present
results show clearly that the shock-capturing capability of the news schemes
are significantly enhanced within a cell-by-cell Adaptive Mesh Refinement
implementation together with time accurate local time stepping (LTS).
