Accuracy-preserving boundary flux quadrature for finite-volume discretization on unstructured grids

In this paper, we derive a general accuracy-preserving boundary flux quadrature formula for the second- and third-order node-centered edge-based finite-volume discretizations on triangular and tetrahedral grids. It is demonstrated that the general formula reduces to some well-known formulas for the second-order scheme in the case of linear fluxes. Some special boundary grids are also examined. In particular, a simple one-point quadrature formula, which is typically used for quadrilateral grids, is shown to be exact for quadratic fluxes on triangular grids with uniformly spaced straight boundaries. Numerical results are presented to demonstrate the accuracy of the general formula, and accuracy deterioration caused by incompatible boundary flux quadrature formulas. In general, the third-order accuracy is lost everywhere in the domain unless the third-order accuracy is maintained at boundary nodes. It is also demonstrated that the third-order scheme does not require high-order curved elements for curved boundaries but requires accurate surface normal vectors defined, which can be estimated by a quadratic interpolation from a given grid, to deliver the designed third-order accuracy.