Modifications to the gradient schemes on unstructured cell centered grids for the accurate determination of gradients near conductivity changes
Marcus Trautmann, Erik Spaniol, Martin Hertel, Uwe Füssel · Physics of Fluids · 2019
Gradient schemes for the cell centered finite volume method on unstructured grids, namely, the divergence theorem and the least squares schemes, have been widely adopted because they have reached a high precision for most applications. These schemes assume continuously differentiable fields for the calculation of the gradients. However, this assumption is violated in the vicinity of conductivity jumps between cells. It is shown that this deficiency leads to a wrong calculation of the gradients and thus the flux density in cells near conductivity changes. For large conductivity jumps, the error of the flux density can exceed several orders of magnitude. Based on theoretical considerations, flux conservative versions of the schemes are derived for the central gradient scheme and extended to the divergence theorem and least squares schemes. The modified schemes named flux conservative divergence theorem and flux conservative least squares take the nonlinearity of a conservation variable near conductivity changes into account and eliminate the error made by the assumption of a continuously differentiable field. The schemes are demonstrated on Cartesian and highly skewed grids with different grid resolutions with a large conductivity jump. The error of the flux density is shown to be reduced by several orders of magnitude up to machine precision for Cartesian grids.