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.
