A novel modified Modica-Mortola equation with a phase-dependent interfacial function

In this study, we propose a novel modified Modica-Mortola (MM) equation with a phase-dependent interfacial function. The classical MM functional has a multiple-well potential and the derived classical MM equation allows multiple local minima. The MM equation has a good property compared to its counterpart vector-valued Allen-Cahn (AC) system because the MM equation consists of a single equation. In the MM equation, one phase-field can represent multiple states. However, with a constant interfacial parameter, the interfacial transition layer is getting wider as the jump across two adjacent phases is higher. To overcome this unwanted phenomenon, we propose a phase-dependent interfacial function which has smaller value if gradient of phase-field is larger. We present several computational experiments to demonstrate the superior performance of the proposed modified MM equation over the conventional MM equation.