  In this paper, a new re-initialization method for the conservative level-set
function is put forward. First, it has been shown that the re-initialization
and advection equations of the conservative level-set function are
mathematically equivalent to the re-initialization and advection equations of
the localized signed distance function. Next, a new discretization for the
spatial derivatives of the conservative level-set function has been proposed.
This new discretization is consistent with the re-initialization procedure and
it guarantees a second-order convergence rate of the interface curvature on
gradually refined grids. The new re-initialization method does not introduce
artificial deformations to stationary and non-stationary interfaces, even when
the number of re-initialization steps is large.
