We investigate the properties of dissipative full discretizations
for the equations of motion associated with
models of flow and radiative transport inside stars. We derive
dissipative space discretizations and demonstrate that together
with specially adapted 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. In particular, for the
computation of semiconvection we can
use partitioned IMEX Runge-Kutta schemes, where the solution of
the implicit part can be reduced to the solution of a
Poisson problem. This yields a significant gain in performance
as compared to popular explicit Runge-Kutta schemes and excels over
more generally applicable fully implicit SDIRK Runge-Kutta methods.