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 $\tilde{z}$ 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 $\tilde{z}$ to be a second-order accurate approximation to $z$ in the max norm is $h$ must be bounded above by $2∕\left(33{\kappa }_{max}\right)$, where ${\kappa }_{max}$ is the maximum magnitude of the curvature $\kappa$ of $z$. This constraint on $h$ is solely in terms of an intrinsic property of the curve $z$, namely ${\kappa }_{max}$, 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 $z-\tilde{z}$ depends linearly on ${\kappa }_{max}$.