  We consider an optimization based limiter for enforcing positivity of
internal energy in a semi-implicit scheme for solving gas dynamics equations.
With Strang splitting, the compressible Navier-Stokes system is splitted into
the compressible Euler equations, solved by the positivity-preserving
Runge-Kutta discontinuous Galerkin (DG) method, and the parabolic subproblem,
solved by Crank-Nicolson method with interior penalty DG method. Such a scheme
is at most second order accurate in time, high order accurate in space,
conservative, and preserves positivity of density. To further enforce the
positivity of internal energy, we impose an optimization based limiter for the
total energy variable to post process DG polynomial cell averages. The
optimization based limiter can be efficiently implemented by the popular first
order convex optimization algorithms such as the Douglas-Rachford splitting
method if using the optimal algorithm parameters. Numerical tests suggest that
the DG method with $\mathbb{Q}^k$ basis and the optimization-based limiter is
robust for demanding low pressure problems such as high speed flows.
