Weak Formulation of Gradient Augmented Level set method to Stephan type problems

Problems associated with dendritic crystallization requires careful implementation of numerical methods. In the past few years various methods have been developed [F.Gibou et al., 2003; F.Gibou and R.Fedkiw, 2005] to track topologically complex, solid-liquid interface in crystal growth simulation. In this work we employ gradient augmented level set method [Nave et al., 2010] to track interface. For maintaining signed distance function we implemented partial differential equation approach from [Anumolu and Trujillo, 2012] and also PhysBAM inbuilt fast marching method, but the results presented here use fast marching method. We discretize the heat equation on a Cartesian grid using implicit backward Euler method and for the nodes neighboring interface, the discretization is performed as proposed in [F.Gibou et al., 2003]. The jump in the first derivatives of the temperature is used to compute interface velocity and Gibbs-Thompson equation is used to compute temperature at the interface. In the following sections, we first describe interface advection methods, followed by heat equation, which is followed by Stephan problem and finally