Cell vertex finite volume discretizations in three dimensions

The cell vertex method is generalized to three dimensions. It is proved that there exists a one‐parameter family of eight‐point three‐dimensional methods with second‐order truncation error on parallelepipeds. Using different triangulations of control volume faces, various finite volume methods are derived. Some of these are identified as members of the aforementioned one‐parameter family and may be regarded as second‐order upwind schemes. A Fourier analysis is used to investigate the spectral properties of these discretizations. Numerical experiments illustrate that second‐order global accuracy is achieved on parallelepiped grids, as suggested by the theory. Randomly perturbed, stretched, sheared meshes are used to test these methods to destruction. It is found that upwinding improves both the accuracy on distorted meshes and the spectrum of the discretization.