On the order of accuracy of the divergence theorem (Green-Gauss) method for calculating the gradient in finite volume methods

Alexandros Syrakos (1) , Stylianos Varchanis (1) , Yannis Dimakopoulos (1) , Apostolos Goulas (2) , John Tsamopoulos (1)

1 Department of Chemical Engineering
2 Department of Mechanical Engineering

The divergence theorem (or Green-Gauss) gradient scheme is among the most popular methods for discretising the gradient operator in second-order accurate finite volume methods, with a long history of successful application in conjunction with structured grids. This together with the ease of application of the scheme on unstructured grids has led to its widespread use in unstructured finite volume methods, which have become dominant in modern engineering applications that involve complex geometrical domains. The present study shows both theoretically and through numerical tests that, unfortunately, on grids of arbitrary skewness, such as produced by most unstructured grid generation algorithms, this scheme is zeroth-order accurate (i.e. it does not converge to the exact gradient operator with grid refinement). Moreover, we use the scheme in the finite volume solution of a simple heat conduction (Poisson equation) problem, with both an in-house code and the popular open-source OpenFOAM solver, and observe that the zeroth-order accuracy of the gradient operator passes on to the finite volume solver as a whole. Second-order accurate results are obtained if a least-squares gradient operator is used instead.

Mots clés: Finite volume method; Green-Gauss gradient; divergence theorem gradient; diffusion equation; OpenFOAM

Domaines: Ingénierie, finance et science [cs.CE]; Analyse numérique [cs.NA]; Mécanique des fluides [physics.class-ph]; Mécanique des matériaux [physics.class-ph]

Alexandros Syrakos, Stylianos Varchanis, Yannis Dimakopoulos, Apostolos Goulas, John Tsamopoulos. On the order of accuracy of the divergence theorem (Green-Gauss) method for calculating the gradient in finite volume methods. 2017. ⟨hal-01532882v1⟩