A unification of least-squares and Green-Gauss gradients under a common projection-based gradient reconstruction framework

TL;DR

This paper introduces a unified framework for gradient reconstruction combining least-squares and Green-Gauss methods using projection techniques, analyzing their stability and performance across various grid types.

Contribution

It unifies least-squares and Green-Gauss gradient schemes through a projection-based approach and provides stability guidelines for high aspect ratio grids.

Findings

Green-Gauss and least-squares gradients are unified under a projection framework.

High aspect ratio grids can cause numerical instability in gradient calculations.

Guidelines are provided to avoid instability in least-squares methods.

Abstract

We propose a family of gradient reconstruction schemes based on the solution of over-determined systems by orthogonal or oblique projections. In the case of orthogonal projections, we retrieve familiar weighted least-squares gradients, but we also propose new direction-weighted variants. On the other hand, using oblique projections that employ cell face normal vectors we derive variations of consistent Green-Gauss gradients, which we call Taylor-Gauss gradients. The gradients are tested and compared on a variety of grids such as structured, locally refined, randomly perturbed, unstructured, and with high aspect ratio. The tests include quadrilateral and triangular grids, and employ both compact and extended stencils, and observations are made about the best choice of gradient and weighting scheme for each case. On high aspect ratio grids, it is found that most gradients can exhibit a kind of numerical instability that may be so severe as to make the gradient unusable. A theoretical analysis of the instability reveals that it is triggered by roundoff errors in the calculation of the cell centroids, but ultimately is due to truncation errors of the gradient reconstruction scheme, rather than roundoff errors. Based on this analysis, we provide guidelines on the range of weights that can be used safely with least squares methods to avoid this instability.