We investigate the properties of dissipative full discretizations
for the advection--diffusion equations associated with models of flow and
radiative transport inside stars. The fundamental equation of motion
is the fully compressible Navier-Stokes equation,
which describes momentum conservation.
The model is completed by a continuity equation
which ensures conservation of mass, and a total energy equation
which describes conservation of the latter.
The equations incorporate a radiative source term Q_{rad}
which is determined as the
stationary limit of the radiative transfer equation.
The equations of hydrodynamics
are closed by the equation of state which describes the relation between the
thermodynamic quantities.
For the spatial discretization of the hyperbolic terms, discretizations of
ENO (essentially non-oscillatory) type are implemented.
These methods use adaptive stencils which are chosen
such as to avoid spurious oscillations in the computed solution.
The spatial derivatives are calculated for each direction separately.
We derive dissipative space discretizations for the parabolic terms and demonstrate that
together with specially adapted implicit total-variation-diminishing
(TVD) Runge-Kutta time discretizations with adaptive step-size
control this yields reliable and efficient integrators for the
underlying high-dimensional nonlinear evolution equations. We
demonstrate that fully implicit SDIRK Runge-Kutta methods enable
large step-sizes as compared to classical explicit integrators and
in conjunction with suitable methods for the associated nonlinear
algebraic equations have the potential to significantly reduce the
computational effort. In the special parameter regime associated
with semiconvection, implicit-explicit methods are demonstrated to
provide the most efficient solution methods. The methods we
recommend based on our analysis provide advantages in efficiency
and accuracy as compared to the classical explicit time integrators.