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Kernels with prescribed surface tension & mobility for threshold dynamics schemes. (English) Zbl 1415.65278

Summary: We show how to construct a convolution kernel that has a desired anisotropic surface tension and desired anisotropic mobility to be used in threshold dynamics schemes for simulating weighted motion by mean curvature of interfaces, including networks of them, in both two and three dimensions. Moreover, we discuss necessary and sufficient conditions for the positivity of the kernel which, in the case of two-phase flow, ensures that the resulting scheme respects a comparison principle and implies convergence to the viscosity solution of the level set formulation of the flow. In particular, we show, in a barrier-type statement, that the kernel cannot possibly be positive unless both the mobility and the surface tension satisfy necessary conditions in three dimensions, and give a complete characterization. Among other results is a threshold dynamics scheme that is guaranteed to dissipate a non-local approximation to the interfacial energy in the fully anisotropic, multiphase setting, using the new kernel construction.

MSC:

65R10 Numerical methods for integral transforms
44A35 Convolution as an integral transform
76T99 Multiphase and multicomponent flows
53A05 Surfaces in Euclidean and related spaces
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