The discontinuous Galerkin (DG) method has been recognized as a promising approach for high-fidelity numerical simulations of turbulent flows (e.g., large-eddy simulation or direct numerical simulation), given its capability to achieve high-order accuracy in complex geometries while damping out spurious high-wavenumber oscillations in advection-dominated regimes. In this work, we analyze an advection-diffusion scheme arising from the strategic use of the Recovery concept to improve the performance of the basic DG spatial discretization on a nearest-neighbors stencil. Following the principle of Recovery, the underlying solution between two neighboring cells is reconstructed in an interface-centered fashion to compute the convective and diffusive fluxes with exceptional accuracy. While certain Recovery-based schemes are known to achieve exceedingly high convergence rates for diffusion (up to $3p+2$ in the cell-average norm, where p is the degree of the polynomial basis), there are more constraining order of accuracy limitations for advection due to the direction of the wind. The key technical challenge we address is to develop an advection-diffusion scheme that is both stable and efficient while leveraging the high accuracy of the Recovery concept on a nearest-neighbors stencil. The usage of Recovery overcomes a fundamental deficiency in typical DG discretizations, namely that a conventional advection-diffusion scheme is reduced from order $2p+1$ accuracy in the advection-dominated regime to order 2p accuracy in the diffusion-dominated regime, as demonstrated by Fourier analysis. Our new Recovery-assisted scheme is instead able to achieve order $2p+2$ accuracy in both the advection-dominated and diffusion-dominated regimes on the nearest-neighbors stencil. By combining the Recovery operator with the mixed formulation, the proposed advection-diffusion scheme achieves improved accuracy compared to established mixed formulation approaches while circumventing the differentiation of the recovered solution, which is a liability in multi-dimensional geometries. Fourier analysis and a suite of linear and nonlinear test problems, including 3D compressible Navier-Stokes, are presented to examine the performance of the new Recovery-assisted scheme; the results show a considerable accuracy advantage compared to a conventional, state-of-the-art DG approach. Furthermore, to simplify the implementation, we show that the Recovery procedure can be recast as a set of derivative-based correction terms, which replicates the Recovery operator on structured meshes while avoiding the traditional complexities of the Recovery operation.

不连续的Galerkin（DG）方法已被公认为是湍流高保真数值模拟（例如大涡模拟或直接数值模拟）的一种有前途的方法，因为它具有在复杂几何体中同时实现阻尼的高阶精度的能力。在对流占主导地位的体制中消除了虚假的高波数振荡。在这项工作中，我们分析了对流扩散方案，该对流扩散方案是由恢复概念的战略性使用所引起的，以改善基本DG空间离散化在最近邻模板上的性能。遵循恢复原理，以接口为中心的方式重构了两个相邻单元之间的基础解决方案，以极高的精度计算对流和扩散通量。$3p+2$在单元平均范数中，其中p是多项式的基数），由于风的方向，对流的精度限制存在更多的约束顺序。我们要解决的关键技术挑战是，开发一种对流扩散方案，该方案既稳定又高效，同时利用最近邻模板上的Recovery概念的高精度。恢复的使用克服了典型DG离散化中的一个基本缺陷，即，常规的对流扩散方案从有序减少$2p+\mathrm{1个}$如傅立叶分析所示，对流占主导地位的精确度达到扩散占主导地位的精确度2 p。相反，我们新的恢复协助计划可以实现订单$2p+2$最近邻模板上以对流为主和以扩散为主的区域的精度。通过将Recovery算子与混合配方相结合，与已建立的混合配方方法相比，所提出的对流扩散方案可提高精度，同时避免了回收溶液的差异化，这在多维几何结构中是一个缺陷。提出了傅里叶分析以及一系列线性和非线性测试问题，包括3D可压缩的Navier-Stokes，以检验新的恢复辅助方案的性能。与常规的最先进的DG方法相比，结果显示了相当大的精度优势。此外，为简化实施，我们证明了可以将恢复过程重铸为一组基于导数的校正项，