On Second- and Higher-Order Finite-Volume Schemes for Viscous Terms on Unstructured Grids

In this paper, we investigate the accuracy of second- and higher-order cell-centered finite-volume schemes for viscous terms. Higher-order nodal-gradient cell-centered finite-volumes are constructed for arbitrary tetrahedral grids based on quadratic and cubic algorithms, but it has been found that the quadratic exactness does not result in third-order accuracy and the cubic exactness leads to fourth-order accuracy. These results have been known for unstructured-grid finite-volume methods, but the mechanism behind such an unexpected behavior is not fully understood. Here, we investigate it in one dimension and report some findings: e.g., both linear and quadratic schemes lead to second-order accuracy on regular grids, and therefore, we cannot expect the quadratic scheme to achieve third-order accuracy on irregular grids. We also derive special parameter values to achieve fourth-order accuracy with the linear and quadratic schemes. Similar discussions are given also to cubic and quartic schemes. The paper concludes with remarks on findings and remaining questions.