A family of first-order accurate gradient schemes for finite volume methods

TL;DR

This paper introduces the Taylor-Gauss gradient scheme for finite volume methods, achieving at least first-order accuracy across various grid types and improving upon existing gradient discretisation techniques.

Contribution

The paper presents a new gradient discretisation scheme, the Taylor-Gauss gradient, that is compatible with second-order finite volume methods and unifies existing gradient approaches.

Findings

Taylor-Gauss gradients are at least first-order accurate on diverse grids.

The scheme compares favorably with existing gradient discretisation methods.

The framework allows for different weighting schemes, enhancing flexibility.

Abstract

A new discretisation scheme for the gradient operator, suitable for use in second-order accurate Finite Volume Methods (FVMs), is proposed. The derivation of this scheme, which we call the Taylor-Gauss (TG) gradient, is similar to that of the least-squares (LS) gradients, whereby the values of the differentiated variable at neighbouring cell centres are expanded in truncated Taylor series about the centre of the current cell, and the resulting equations are summed after being weighted by chosen vectors. Unlike in the LS gradients, the TG gradients use vectors aligned with the face normals, resembling the Green-Gauss (GG) gradients in this respect. Thus, the TG and LS gradients belong in a general unified framework, within which other gradients can also be derived. The similarity with the LS gradients allows us to try different weighting schemes (magnitudes of the weighting vectors) such as weighting by inverse distance or face area. The TG gradients are tested on a variety of grids such as structured, locally refined, randomly perturbed, and with high aspect ratio. They are shown to be at least first-order accurate in all cases, and are thus suitable for use in second-order accurate FVMs. In many cases they compare favourably over existing schemes.