We describe a new and simple strategy based on the Gauss divergence theorem for obtaining centroidal gradients on unstructured meshes. Unlike the standard Green–Gauss (SGG) reconstruction which requires face values of quantities whose gradients are sought, the proposed approach reconstructs the gradients using the normal derivative(s) at the faces. The new strategy, referred to as the Modified Green–Gauss (MGG) reconstruction results in consistent gradients which are at least first-order accurate on arbitrary polygonal meshes. We show that the MGG reconstruction is linearity preserving independent of the mesh topology and retains the consistent behaviour of gradients even on meshes with large curvature and high aspect ratios. The gradient accuracy in MGG reconstruction depends on the accuracy of discretisation of the normal derivatives at faces and this necessitates an iterative approach for gradient computation on non-orthogonal meshes. Numerical studies on different mesh topologies demonstrate that MGG reconstruction gives accurate and consistent gradients on non-orthogonal meshes, with the number of iterations proportional to the extent of non-orthogonality. The MGG reconstruction is found to be consistent even on meshes with large aspect ratio and curvature with the errors being lesser than those from linear least-squares reconstruction. A non-iterative strategy in conjunction with MGG reconstruction is proposed for gradient computations in finite volume simulations that achieves the accuracy and robustness of MGG reconstruction at a cost equivalent to that of SGG reconstruction. The efficacy of this strategy for fluid flow problems is demonstrated through numerical investigations in both incompressible and compressible regimes. The MGG reconstruction may, therefore, be viewed as a novel and promising blend of least-squares and Green–Gauss based approaches which can be implemented with little effort in open-source finite-volume solvers and legacy codes.

我们描述了一种基于高斯散度定理的新的简单策略，用于获得非结构化网格上的质心梯度。与标准的Green-Gauss（SGG）重建方法要求的面值是要寻找其梯度的量不同，所提出的方法使用面的法线导数来重建梯度。这种新的策略被称为修正格林-高斯（MGG）重建，可产生一致的梯度，该梯度在任意多边形网格上至少具有一阶精度。我们表明，MGG重构是独立于网格拓扑的线性保留，即使在具有大曲率和高长宽比的网格上，也可以保持渐变的一致行为。MGG重建中的梯度精度取决于面上法线导数离散化的精度，因此有必要采用迭代方法对非正交网格进行梯度计算。对不同网格拓扑的数值研究表明，MGG重建可在非正交网格上提供准确且一致的渐变，且迭代次数与非正交程度成正比。发现即使在长宽比和曲率较大的网格上，MGG重建也是一致的，其误差小于线性最小二乘重建的误差。提出了一种与MGG重建相结合的非迭代策略，用于有限体积模拟中的梯度计算，从而以与SGG重建相同的成本实现了MGG重建的准确性和鲁棒性。通过在不可压缩和可压缩状态下进行的数值研究证明了该策略对流体流动问题的有效性。因此，MGG重构可以被视为一种新颖且有前途的最小二乘和基于Green-Gauss的方法的混合，可以在开源有限体积求解器和遗留代码中轻松实现。