Stability Analysis of the Reconstruction Step of the Voronoi Implicit Interface Method

In this paper we analyze the reconstruction step of the Voronoi implicit interface method (VIIM) introduced by Saye and Sethian. The VIIM is a powerful method to track multiple interfaces with a single function. The central idea of the VIIM is to use an unsigned distance function to represent the multiphase system, unlike the level set method which uses a signed distance function. The unsigned distance function is evolved according to the physics of the problem as in the level set method, but the new multiphase interface is defined in the so-called reconstruction step as the Voronoi interface of the updated $\varepsilon$-level lines. In this paper we analyze this reconstruction step in two dimensions. We show several geometric properties of the resulting interface. Most importantly, we study the asymptotic behavior of the interface when the reconstruction step is iterated. For two phases, we show that the process converges for any initial configuration and that the curvature of the limit phases are bounde...

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