Gradient Calculation Methods on Arbitrary Polyhedral Unstructured Meshes for Cell-Centered CFD Solvers

Abstract

A survey of gradient reconstruction methods for cell-centered data on unstructured meshes is conducted within the scope of accuracy assessment. Formal order of accuracy, as well as error magnitudes for each of the studied methods, are evaluated on a complex mesh of various cell types through consecutive local scaling of an analytical test function. The tests highlighted several gradient operator choices that can consistently achieve 1 st order accuracy regardless of cell type and shape.The tests further offered error comparisons for given cell types, leading to the observation that the "ideal" gradient operator choice is not universal. Practical implications of the results are explored via CFD solutions of a 2D inviscid standing vortex, portraying the discretization error properties. A relatively naive, yet largely unexplored, approach of local curvilinear stencil transformation exhibited surprisingly favorable properties.

A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods

“…In the literature, usually the gradient discretisation is only briefly discussed within an overall presentation of a FVM, with only a relatively limited number of studies devoted specifically to it (e.g. [6,[16][17][18][19][20]). This suggests that existing gradient schemes are deemed satisfactory, and in fact there seems to be a widespread misconception that the DT and LS schemes are second-order accurate on any type of grid.…”

“…5.4, concludes that this is due to grid skewness. Later, Sozer et al [20] tested a simpler variant of the DT gradient that uses arithmetic averaging instead of linear interpolation and proved that in the one-dimensional case it converges to incorrect values if the grid is not uniform. In numerical tests they also noticed that the scheme is inconsistent on two-dimensional grids of arbitrary topology.…”

Analysis of the Numerical Diffusion in Anisotropic Mediums: Benchmarks for Magnetic Field Aligned Meshes in Space Propulsion Simulations

“…Both Diskin [29] and Sozer [30] review the use of the Green-Gauss (GG) and the LSQR (both Weighted -WLSQR-and Unweighted -ULSQR-) methods in terms of order of accuracy for GR; Sozer also performs analysis on additional methods, showcasing that, while consistency in the GG and LSQR methods might vary with mesh regularity, other methods tend to have lower orders of accuracy. Recently several authors have developed hybrid Green-Gauss/Weighted-Least-Squares methods [30,31], which might offer the best compromise between GR methods.…”

“…Recently several authors have developed hybrid Green-Gauss/Weighted-Least-Squares methods [30,31], which might offer the best compromise between GR methods. However, for the sake of simplicity and since the objective here is to provide an insight to the issues arising in the GR due to low mesh quality, the following analysis is performed using a WLSQR method.…”

“…In particular, a method equivalent to the WLSQR Face Interpolation -WLSQRFI-method described by Sozer [30] has been implemented because of the selection of a cell-centered FVM discretization, that requires gradients at the element facets. The method is based on the Taylor series expansion.…”