On reinitializing level set functions

In this paper, we consider reinitializing level functions through equation ϕt+sgn(ϕ0)(‖∇ϕ‖-1)=0ϕt+sgn(ϕ0)(‖∇ϕ‖-1)=0[16]. The method of Russo and Smereka [11] is taken in the spatial discretization of the equation. The spatial discretization is, simply speaking, the second order ENO finite difference with subcell resolution near the interface. Our main interest is on the temporal discretization of the equation. We compare the three temporal discretizations: the second order Runge–Kutta method, the forward Euler method, and a Gauss–Seidel iteration of the forward Euler method. The fact that the time in the equation is fictitious makes a hypothesis that all the temporal discretizations result in the same result in their stationary states. The fact that the absolute stability region of the forward Euler method is not wide enough to include all the eigenvalues of the linearized semi-discrete system of the second order ENO spatial discretization makes another hypothesis that the forward Euler temporal discretization should invoke numerical instability. Our results in this paper contradict both the hypotheses. The Runge–Kutta and Gauss–Seidel methods obtain the second order accuracy, and the forward Euler method converges with order between one and two. Examining all their properties, we conclude that the Gauss–Seidel method is the best among the three. Compared to the Runge–Kutta, it is twice faster and requires memory two times less with the same accuracy.