Green-Gauss Theorem in FV gives the accuracy of a Central Finite Difference Equation in calculating Diffusive Flux or gradient of a quantity across a face or boundary of two grids. You can try it by using a simple rectangular grid having different temperatures and that would lead to the same answer as CD. But numerically investigating the error with a practical solution could lead to a first order accurate result. you can plot the log of error with respect to log of grid size and the variation would give you the accuracy order.

Thanks for ur advice! Ur knowledge about accuracy order measuring is what i am doing right now.

If u r interested, the attached paper mentioned the accuracy degradation problem of GG methods, the authors proved that GG method with simple averaging  in cell gradient reconstruction may fail to reproduce gradient of a linear function exactly via one-dimensional derivation. 

Conference Paper Gradient Calculation Methods on Arbitrary Polyhedral Unstruc...

Hi Nianhua

Are you interested by gradient reconstruction on unstrutured meshes ?

In this cas it depends on the formulation you used : cell centered or node centerd.  Mesh distorsion is an important factor .and the gradient approxiamtion method used (Green-Gauss, Least-Square Weghted Least-Square etc.

Some interested reference that can help you are :

Accuracy of Gradient Reconstruction on Grids with High Aspect Ratio

NIA Report No. 2008-12  ,2008.

Boris Diskin and James Thomas

Numerical aspects of computing high-Reynolds number flow on unstructured

meshes, AIAA Paper 91-0721, 29-th AIAA Aerospace Science Meeting, Reno, NV, 

1991.

T.J Barth

Revisiting the Least-Squares Procedure for Gradient Reconstruction on Unstructured

Meshes.

AIAA Paper 2003-3986, 18-th AIAA CFD conference, Orlando, FL, June 2003.

Dimitri J. Mavriplis

Green–Gauss/Weighted-Least-Squares Hybrid Gradient Reconstruction for Arbitrary Polyhedra Unstructured Grids. AIAA Journal, Vol. 51, No. 11 (2013),

Eiji Shima*Keiichi Kitamura†Takanori Haga

Recent developments in high order K-exact reconstruction on unstructured meshes

AIAA-93-0668, 1993

T. Barth

The following Book and thesis :

Computational Fluid Dynamics: Principles and Applications, Elsevier by Jiri Blazek 

Efficient Implementation of High-Order Accurate Numerical Methods on Unstructured Grids by Wanai Li 

Doctoral Thesis accepted by Tsinghua University, Beijing, 2014.

Best regards 

P. Galon

http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20090007493.pdf

http://www.ewp.rpi.edu/hartford/~ernesto/F2013/EP/MaterialsforStudents/Hiester/Mavriplis2003-LeastSquareGradient-aiaa20033986.pdf

generally, using unstructured grid the best way to produce high accuracy order in FV is to create a mix with Finte Element-based second degree shape functions. For example, this method is reported in the book of Peric & Ferziger.

I tried to develop such method some years ago

https://www.researchgate.net/publication/229470711_2-D_Transmitral_flows_simulation_by_means_of_the_immersed_boundary_method_on_unstructured_grids

Article 2-D Transmitral flows simulation by means of the immersed bo...