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We put forward the use of total-variation-diminishing (or more generally,strong stability preserving) implicit-explicit Runge-Kutta methods for the timeintegration of the equations of motion associated with the semiconvectionproblem in the simulation of stellar convection. The fully compressibleNavier-Stokes equation, augmented by continuity and total energy equations, andan equation of state describing the relation between the thermodynamicquantities, is semi-discretized in space by essentially non-oscillatory schemesand dissipative finite difference methods. It is subsequently integrated intime by Runge-Kutta methods which are constructed such as to preserve the totalvariation diminishing (or strong stability) property satisfied by the spatialdiscretization coupled with the forward Euler method. We analyse the stability,accuracy and dissipativity of the time integrators and demonstrate that themost successful methods yield a substantial gain in computational efficiency ascompared to classical explicit Runge-Kutta methods.

Total-Variation-Diminishing Implicit-Explicit Runge-Kutta Methods for the Simulation of Double-Diffusive Convection in Astrophysics