A variational lagrangian scheme for a phase-field model: A discrete energetic variational approach

In this paper, we propose a variational Lagrangian scheme for a modified phase-field model, which can compute the equilibrium states of the original Allen-Cahn type model. Our discretization is based on a prescribed energy-dissipation law in terms of the flow map. By employing a discrete energetic variational approach, this scheme preserves the variational structure of the continuous energy-dissipation law and is energy stable. Plentiful numerical tests show that, by choosing the initial value properly, our method can compute the desired equilibrium state and capture the thin diffuse interface with a small number of mesh points.