An optimization based limiter for enforcing positivity in a semi-implicit discontinuous Galerkin scheme for compressible Navier-Stokes equations

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.