Given a two times differentiable curve in the plane, I prove that — using only the volume fractions associated with the curve — one can construct a piecewise linear approximation that is second-order in the max norm. I derive two parameters that depend only on the grid size and the curvature of the curve, respectively. When the maximum curvature in the 3 by 3 block of cells centered on a cell through which the curve passes is less than the first parameter, the approximation in that cell will be second-order. Conversely, if the grid size in this block is greater than the second parameter, the approximation in the center cell can be less than second-order. Thus, this parameter provides an a priori test for when the interface is under-resolved, so that when the interface reconstruction method is coupled to an adaptive mesh refinement algorithm, this parameter may be used to determine when to locally increase the resolution of the grid.
"On the second-order accuracy of volume-of-fluid interface reconstruction algorithms: convergence in the max norm." Commun. Appl. Math. Comput. Sci. 5 (1) 99 - 148, 2010. https://doi.org/10.2140/camcos.2010.5.99