On the second-order accuracy of volume-of-fluid interface reconstruction algorithms: convergence in the max norm

Abstract

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.