  Finite volume methods (FVMs) constitute a popular class of methods for the
numerical simulation of fluid flows. Among the various components of these
methods, the discretisation of the gradient operator has received less
attention despite its fundamental importance with regards to the accuracy of
the FVM. The most popular gradient schemes are the divergence theorem (DT) (or
Green-Gauss) scheme, and the least-squares (LS) scheme. Both are widely
believed to be second-order accurate, but the present study shows that in fact
the common variant of the DT gradient is second-order accurate only on
structured meshes whereas it is zeroth-order accurate on general unstructured
meshes, and the LS gradient is second-order and first-order accurate,
respectively. This is explained through a theoretical analysis and is confirmed
by numerical tests. The schemes are then used within a FVM to solve a simple
diffusion equation on unstructured grids generated by several methods; the
results reveal that the zeroth-order accuracy of the DT gradient is inherited
by the FVM as a whole, and the discretisation error does not decrease with grid
refinement. On the other hand, use of the LS gradient leads to second-order
accurate results, as does the use of alternative, consistent, DT gradient
schemes, including a new iterative scheme that makes the common DT gradient
consistent at almost no extra cost. The numerical tests are performed using
both an in-house code and the popular public domain PDE solver OpenFOAM.
