In this paper, we are interested in constructing a scheme solving compressible Navier--Stokes equations, with desired properties including high order spatial accuracy, conservation, and positivity-preserving of density and internal energy under a standard hyperbolic type CFL constraint on the time step size, e.g., $\Delta t=\mathcal O(\Delta x)$. Strang splitting is used to approximate convection and diffusion operators separately. For the convection part, i.e., the compressible Euler equation, the high order accurate postivity-preserving Runge--Kutta discontinuous Galerkin method can be used. For the diffusion part, the equation of internal energy instead of the total energy is considered, and a first order semi-implicit time discretization is used for the ease of achieving positivity. A suitable interior penalty discontinuous Galerkin method for the stress tensor can ensure the conservation of momentum and total energy for any high order polynomial basis. In particular, positivity can be proven with $\Delta t=\mathcal{O}(\Delta x)$ if the Laplacian operator of internal energy is approximated by the $\mathbb{Q}^k$ spectral element method with $k=1,2,3$. So the full scheme with $\mathbb{Q}^k$ ($k=1,2,3$) basis is conservative and positivity-preserving with $\Delta t=\mathcal{O}(\Delta x)$, which is robust for demanding problems such as solutions with low density and low pressure induced by high-speed shock diffraction. Even though the full scheme is only first order accurate in time, numerical tests indicate that higher order polynomial basis produces much better numerical solutions, e.g., better resolution for capturing the roll-ups during shock reflection.

翻译：本文旨在构造一种求解可压缩Navier-Stokes方程的数值格式，该格式具备高阶空间精度、守恒性以及在标准双曲型CFL时间步长约束（如$\Delta t=\mathcal O(\Delta x)$）下保持密度和内能正性等理想性质。采用Strang分裂方法分别逼近对流和扩散算子。对于对流部分（即可压缩Euler方程），可使用高阶保正Runge-Kutta间断Galerkin方法；对于扩散部分，考虑内能方程而非总能方程，并采用一阶半隐式时间离散以简化保正实现。采用适用于应力张量的内罚间断Galerkin方法，可在任意高阶多项式基下保证动量和总能守恒。特别地，当内能的Laplacian算子由$k=1,2,3$阶$\mathbb{Q}^k$谱元法逼近时，可证明在$\Delta t=\mathcal{O}(\Delta x)$条件下满足保正性。因此，采用$\mathbb{Q}^k$（$k=1,2,3$）基的完整格式在$\Delta t=\mathcal{O}(\Delta x)$条件下具有守恒性和保正性，对于高速激波衍射引起的低密度低压等苛刻问题具有良好的鲁棒性。尽管完整格式仅具有时间一阶精度，数值实验表明，采用高阶多项式基可显著改善数值解质量，例如在激波反射过程中能更清晰地捕捉卷起涡结构。