Abstract:

The core contribution of this thesis is the establishment of a precise connection between median filter (sorting-based) level set schemes and threshold dynamics. This connection facilitates the development of level set methods informed by recent advances in threshold dynamics, enabling simulation of the mean curvature motion of interface networks under varying surface tension and mobility conditions. We then extend these median filters to an anisotropic wetting/dewetting scenario where one of the three phases remains stationary while triple junctions form at their intersections. This scenario serves as a testbed for examining complex triple junction conditions due to anisotropy. Numerical evidence supports the correct angle condition at these junctions, encouraging further exploration into fully anisotropic multiphase flow dynamics.

Additionally, building on this connection between level set methods and threshold dynamics, we develop a median filter scheme that is second-order accurate in time and monotone, ensuring convergence to viscosity solutions within established theoretical frameworks.