This object provides a second-order, Total Variation Diminishing (TVD)-style scheme for interpolating a cell-centered advected quantity to a finite-volume face on unstructured grids. It is formulated as a limited blending weight between upwind and linear (geometric) interpolation, so it can be assembled using only two-cell (elem/neighbor) matrix weights (Moukalled et al. (2016), Jasak (1996), Greenshields and Weller (2022), Harten (1997)).

Let and denote the upwind and downwind cell-centered values on a face (as determined by the sign of the face mass flux). The face value is written in a Normalized Variable Diagram (NVD) / Greenshields-type blending form (Greenshields and Weller (2022), Jasak (1996)):

where controls how much downwind information is admitted. This method computes

with:

• the user scaling factor ("blending_factor"). gives pure upwind; gives the full limited blending. Values can be useful for improving robustness of fully implicit solves.

• the geometric linear-interpolation weight associated with the upwind cell. On a uniform orthogonal mesh, and the linear scheme is recovered when .

• the limiter coefficient computed from a smoothness indicator that compares upwind and downwind variation.

When "limit_to_linear" is enabled (default), the blending is additionally constrained so the scheme is never more downwind-biased than linear interpolation:

For unstructured grids, is typically formed using a virtual upwind state built from the upwind cell gradient and the vector connecting neighboring cell centroids (see Moukalled et al. (2016)). In smooth regions (), and the method approaches linear interpolation. At local extrema or discontinuities (), and the method reverts to upwind, suppressing non-physical oscillations.