Threshold dynamics type approximation schemes for propagating fronts

Abstract

We study the convergence of general threshold dynamics type approx- imation schemes to hypersurfaces moving with normal velocity depending on the normal direction and the curvature tensor. We also present results about the asymptotic shape of fronts propagating by threshold dynamics. Our results generalize and extend models introduced in the theories of cellular automaton and motion by mean curvature. In this paper we study the convergence of general threshold dynamics type ap- proximation schemes to hypersurfaces moving with normal velocity depending on the normal direction and the curvature tensor. These schemes are generalizations and extensions of the threshold dynamics models introduced by Gravner and Grieath (GrGr) to study cellular automaton modeling of excitable media and by Bence, Merriman and Osher (BMO) to study the mean curvature evolution. Cellular automaton models are mathematical models used to understand the transmission of periodic waves through environments such as a network or a tissue. A common feature of many such models is that some threshold level of excitation must occur in a neighborhood of a location to become excited and conduct a pulse. Typical physical systems which exhibit such phenomenology are, among others, neural networks, cardiac muscle, Belousov-Zhabotinsky oscillating chemical reaction, etc. Interfaces (fronts, hypersurfaces) in R N evolving with normal velocity V ¼ vðDn;nÞ; ð0:1Þ