Stabilized conservative level set method

This paper addresses one of the main challenges of the conservative level set method, namely the ill-conditioned behavior of the normal vector away from the interface. An alternative formulation for reconstruction of the interface is proposed. Unlike commonly used methods, which rely on a unit normal vector, the Stabilized Conservative Level Set (SCLS) makes use of a modified normal vector with diminishing magnitude away from the interface. With the new formulation, in the vicinity of the interface the reinitialization procedure utilizes compressive flux and diffusive terms only in normal direction with respect to the interface, thus, preserving the conservative level set properties, while away from the interface the directional diffusion mechanism automatically switches to homogeneous diffusion. The proposed formulation is robust and general. It is especially well suited for use with the adaptive mesh refinement (AMR) approaches, since for computational accuracy high resolution is only required in the vicinity of the interface, while away from the interface low resolution simulations might be sufficient. All of the results reported in this paper are obtained using the Adaptive Wavelet Collocation Method, a general arbitrary order AMR-type method, which utilizes wavelet decomposition to adapt on steep gradients in the solution while retaining a predetermined order of accuracy. Numerical solution for a number of benchmark problems has been carried out to demonstrate the performance of the SCLS method.