In this article we present a high order cell-centered numerical scheme in space and time for the solution of the compressible Navier-Stokes equations. To deal with multiscale phenomena induced by the different speeds of acoustic and material waves, we propose a semi-implicit time discretization which allows the CFL-stability condition to be independent of the fast sound speed, hence improving the efficiency of the solver. This is particularly well suited for applications in the low Mach regime with a rather small fluid velocity, where the governing equations tend to the incompressible model. The momentum equation is inserted into the energy equation yielding an elliptic equation on the pressure. The class of implicit-explicit (IMEX) time integrators is then used to ensure asymptotic preserving properties of the numerical method and to improve time accuracy. High order in space is achieved relying on implicit finite difference and explicit CWENO reconstruction operators, that ultimately lead to a fully quadrature-free scheme. To relax the severe parabolic restriction on the maximum admissible time step related to viscous contributions, a novel implicit discretization of the diffusive terms is designed. A variational approach based on the discontinuous Galerkin (DG) spatial discretization is devised in order to obtain a discrete cell-centered Laplace operator. High order corner gradients of the velocity field are employed in 3D to derive the Laplace discretization, and the resulting viscous system is proven to be symmetric and positive definite. As such, it can be conveniently solved at the aid of the conjugate gradient method. Numerical results confirm the accuracy and the robustness of the novel schemes in the challenging stiff limit of the governing equations characterized by low Mach numbers.

在本文中，我们提出了一种高阶单元中心数值格式，用于求解可压缩的 Navier-Stokes 方程。为了处理由声波和物质波的不同速度引起的多尺度现象，我们提出了一种半隐式时间离散化，它允许 CFL 稳定条件独立于快声速，从而提高求解器的效率。这特别适用于流体速度相当小的低马赫状态下的应用，其中控制方程倾向于不可压缩模型。动量方程被插入到能量方程中，得到一个关于压力的椭圆方程。然后使用隐式-显式 (IMEX) 时间积分器类来确保数值方法的渐近保持特性并提高时间精度。空间中的高阶是依靠隐式有限差分和显式 CWENO 重构算子实现的，最终导致完全无正交方案。为了放松对与粘性贡献相关的最大允许时间步长的严格抛物线限制，设计了一种新颖的扩散项隐式离散化。设计了一种基于不连续伽辽金（DG）空间离散化的变分方法，以获得离散的以单元为中心的拉普拉斯算子。在 3D 中采用速度场的高阶角梯度来推导拉普拉斯离散化，并且由此产生的粘性系统被证明是对称的和正定的。因此，可以借助共轭梯度法方便地求解。数值结果证实了新方案在具有挑战性的以低马赫数为特征的控制方程的刚性极限中的准确性和鲁棒性。