An Algorithm to Detect High-Γ Regions for Three Dimensional Unstructured Grids

In this paper, we propose an algorithm to estimate a parameter Γ, which is a measure of a combined effect of aspect ratio and curvature associated with a local stencil (e.g., a least-squares stencil), for three- dimensional unstructured grids. Stencils with a large Γ are known to cause various troubles in gradient accuracy and iterative convergence of computational-fluid-dynamics solvers. Therefore, it can be helpful to detect such stencils, so that we can overcome such difficulties by locally modifying algorithms in the discretization, e.g., least-squares stencil construction algorithms (or by modifying the grid). To estimate Γ for a given stencil, we first compute the eigenvalues and eigenvectors of a covariance matrix of the stencil and use the eigenvector corresponding to the minimum-magnitude eigenvalue as an estimate of a small-grid- spacing direction. Then, we find the minimum and maximum distances between the center node/cell and neighbor nodes/cells projected along the small-grid-spacing direction, and estimate Γ by their ratio. In this work, we discuss its application as an adaptive least-squares gradient stencil augmentation algorithm for a nodal-gradient-based cell-centered finite-volume solver: apply a stencil augmentation in high-Γ regions for robustness and otherwise keep the original stencils.