We present a generic framework for gradient reconstruction schemes on unstructured meshes using the notion of a dyadic sum-vector product. The proposed formulation reconstructs centroidal gradients of a scalar from its directional derivatives along specific directions in a suitably defined neighbourhood. We show that existing gradient reconstruction schemes can be encompassed within this framework by a suitable choice of the geometric vectors that define the dyadic sum tensor. The proposed framework also allows us to re-interpret certain hybrid schemes, which might not be derivable through traditional routes. Additionally, a generalization of flexible gradient schemes is proposed that can be employed to enhance the robustness of consistent gradient schemes without compromising on the accuracy of the computed gradients.

翻译：本文提出一个通用框架，用于非结构网格上的梯度重建方案，该框架基于并矢求和向量积的概念。所提出的表述在适当定义的邻域内，通过标量沿特定方向的方向导数重建其质心梯度。我们证明，通过适当选择定义并矢求和张量的几何向量，现有的梯度重建方案均可纳入此框架。该框架还允许我们重新解释某些无法通过传统途径导出的混合方案。此外，本文提出了一种灵活梯度方案的推广形式，该方案可在不降低计算梯度精度的前提下增强一致梯度方案的鲁棒性。