Second-order accuracy of volume-of-fluid interface reconstruction algorithms ii: An improved constraint on the cell size

Abstract

In a previous article in this journal the author proved that, given a square grid of side h covering a two times continuously differentiable simple closed curve z in the plane, one can construct a pointwise second-order accurate piecewise linear approximation z̃ to z from just the volume fractions due to z in the grid cells. In the present article the author proves a sufficient condition for z̃ to be a second-order accurate approximation to z in the max norm is h must be bounded above by 2/ (33κmax) where κmax is the maximum magnitude of the curvature κ. of z. This constraint on h is solely in terms of an intrinsic property of the curve z, namely κmax, which is invariant under rotations and translations of the grid. It is also far less restrictive than the constraint presented in the previous article. An important consequence of the proof in the present article is that the max norm of the difference z- z̃ Qz depends linearly on κmax. © 2013 Mathematical Sciences Publishers.