Differential equation based constrained reinitialization for level set methods

A partial differential equation based reinitialization method is presented in the framework of a localized level set method. Two formulations of the new reinitialization scheme are derived. These formulations are modifications of the partial differential equation introduced by Sussman et al. [M. Sussman, P. Smereka, S. Osher, A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys. 114 (1994) 146-159] and, in particular, improvements of the second-order accurate modification proposed by Russo and Smereka [G. Russo, P. Smereka, A remark on computing distance functions, J. Comput. Phys. 163 (2000) 51-67]. The first formulation uses the least-squares method to explicitly minimize the displacement of the zero level set within the reinitialization. The overdetermined problem, which is solved in the first formulation of the new reinitialization scheme, is reduced to a determined problem in another formulation such that the location of the interface is locally preserved within the reinitialization. The second formulation is derived by systematically minimizing the number of constraints imposed on the reinitialization scheme. For both systems, the resulting algorithms are formulated in a three-dimensional frame of reference and are remarkably simple and efficient. The new formulations are second-order accurate at the interface when the reinitialization equation is solved with a first-order upwind scheme and do not diminish the accuracy of high-order discretizations of the level set equation. The computational work required for all components of the localized level set method scales with O(N). Detailed analyses of numerical solutions obtained with different discretization schemes evidence the enhanced accuracy and the stability of the proposed method, which can be used for localized and global level set methods.