  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.
