In this paper, we propose a flux correction technique generally applicable to practical finite-volume discretizations of a single flux evaluation per face for achieving second-order accuracy on arbitrary polyhedral grids involving non-planar faces. The proposed technique is derived from the k-exact finite-volume discretization approach originally introduced by Brenner. We take it as a general methodology for constructing a second-order discretization on arbitrary polyhedral grids, and identify a term considered as missing in other practical finite-volume discretizations; it is thus considered as a reason for the loss of second-order accuracy in many of such methods: e.g., a cell-centered discretization loses second-order accuracy on grids with non-planar faces as typical in polyhedral grids. In particular, we write the term as a vector involving flux gradients and show that it can be easily implemented as a correction to a numerical flux at each face. Also, we will show that a consistent definition of a control volume is critically important for achieving second-order accuracy on general polyhedral grids. We then discuss its general applicability and impact on accuracy and efficiency with simple but illustrative examples: the edge-based solver is made to achieve faster iterative convergence on irregular tetrahedral grids with adjustable centroids while preserving second-order accuracy; both the edge-based and cell-centered discretizations are made second-order accurate on irregular prismatic grids having non-planar faces.

在本文中，我们提出了一种通量校正技术，该技术通常适用于每个面的单个通量评估的实际有限体积离散化，以在涉及非平面的任意多面体网格上实现二阶精度。所提出的技术源自k- 最初由 Brenner 引入的精确有限体积离散化方法。我们将其作为在任意多面体网格上构建二阶离散化的通用方法，并确定在其他实际有限体积离散化中被认为缺失的项；因此，这被认为是许多此类方法中二阶精度损失的原因：例如，在多面体网格中典型的非平面网格上，以单元为中心的离散化会损失二阶精度。特别是，我们将这个术语写成一个涉及通量梯度的向量，并表明它可以很容易地实现为对每个面的数值通量的校正。此外，我们将证明控制体积的一致定义对于在一般多面体网格上实现二阶精度至关重要。然后，我们通过简单但说明性的示例讨论其一般适用性以及对精度和效率的影响：基于边缘的求解器旨在在具有可调节质心的不规则四面体网格上实现更快的迭代收敛，同时保持二阶精度；在具有非平面的不规则棱柱网格上，基于边缘和以单元为中心的离散化都是二阶精度的。