  We put forward the use of total-variation-diminishing (or more generally,
strong stability preserving) implicit-explicit Runge-Kutta methods for the time
integration of the equations of motion associated with the semiconvection
problem in the simulation of stellar convection. The fully compressible
Navier-Stokes equation, augmented by continuity and total energy equations, and
an equation of state describing the relation between the thermodynamic
quantities, is semi-discretized in space by essentially non-oscillatory schemes
and dissipative finite difference methods. It is subsequently integrated in
time by Runge-Kutta methods which are constructed such as to preserve the total
variation diminishing (or strong stability) property satisfied by the spatial
discretization coupled with the forward Euler method. We analyse the stability,
accuracy and dissipativity of the time integrators and demonstrate that the
most successful methods yield a substantial gain in computational efficiency as
compared to classical explicit Runge-Kutta methods.
