Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations: Viscous Fluxes

Finite-volume discretization schemes for viscous fluxes on general grids are compared using node-centered and cell-centered approaches. The grids range from regular grids to highly irregular grids, including random perturbations of the grid nodes. Accuracy and complexity are studied for four nominally second-order accurate schemes: a node-centered scheme and three cell-centered schemes (a node-averaging scheme and two schemes using least-squares face-gradient reconstruction). The two least-squares schemes use either a nearest-neighbor or an adaptive-compact stencil at a face. The node-centered and least-squares schemes have similarly low levels of complexity. The node-averaging scheme has the highest complexity and can fail to converge to the exact solution when clipping of the node-averaged values is used. On highly anisotropic grids, typical of those encountered in grid adaptation, the least-squares schemes, the node-averaging scheme without clipping, and the node-centered scheme demonstrate similar second-order accuracies per degree of freedom. On anisotropic grids over a curved body, typical of turbulent flow simulations, the node-centered scheme is second-order accurate. The node-averaging scheme may degenerate on mixed-element grids. The least-squares schemes have to be amended to maintain second-order accuracy by either introducing a local approximate mapping or modifying the stencil to reflect the direction of strong coupling. Overall, the accuracies of the node-centered and the best cell-centered schemes are comparable at an equivalent number of degrees of freedom on isotropic and curved anisotropic grids. On stretched, randomly perturbed grids in a rectangular geometry, both gradient and discretization errors for all schemes are orders of magnitude higher than corresponding errors on regular grids.