A consistent solution of the re-initialization equation in the conservative level-set method

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.