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.

翻译：我们提出一种基于优化的限制器，用于在半隐式格式求解气体动力学方程时保持内能的正性。通过Strang分裂，可压缩Navier-Stokes系统被分解为可压缩欧拉方程（采用保正性龙格-库塔间断伽辽金方法求解）和抛物性子问题（采用Crank-Nicolson方法结合内罚间断伽辽金方法求解）。该格式在时间上具有二阶精度，在空间上具有高阶精度，满足保守性，并能保持密度正性。为进一步强制保持内能正性，我们对总能量变量施加基于优化的限制器，以对间断伽辽金单元多项式平均进行后处理。若采用最优算法参数，该优化限制器可通过流行的凸优化一阶算法（如Douglas-Rachford分裂方法）高效实现。数值实验表明，采用$\mathbb{Q}^k$基函数和基于优化的限制器的间断伽辽金方法，在高速流动等低压问题中具有鲁棒性。