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，该格式需具备高阶空间精度、守恒性以及密度和内能在标准双曲型CFL时间步长（例如$\Delta t=\mathcal O(\Delta x)$）约束下的保正性。采用Strang分裂分别近似对流算子和扩散算子。对于对流部分（即可压缩Euler方程），可使用高阶精确保正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性实现。通过适用于应力张量的内罚间断Galerkin方法，可确保任意高阶多项式基下的动量与总能守恒性。特别地，当内能拉普拉斯算子采用$\mathbb{Q}^k$（$k=1,2,3$）谱元法近似时，可证明在$\Delta t=\mathcal{O}(\Delta x)$条件下保正。因此，采用$\mathbb{Q}^k$（$k=1,2,3$）基的完整格式在$\Delta t=\mathcal{O}(\Delta x)$下具有守恒性和保正性，对高速激波衍射导致的低密度低压问题等严苛工况具有稳健性。尽管完整格式仅具备一阶时间精度，数值试验表明高阶多项式基可显著提升数值解质量，例如激波反射过程中对卷起结构的更高分辨率捕捉。