An Unconditionally Stable Scheme for the Modica-Mortola Equation with a Variable Interfacial Parameter

Abstract

In this paper, we present an unconditionally stable numerical scheme for the modified Modica-Mortola (MM) equation with a variable interfacial parameter. The original MM equation with a constant interfacial parameter is derived from a functional having a multiple-well potential. The MM model has a good characteristic that a single function can effectively represent an unlimited number of phases. However, a drawback arises in that the interfacial transition layer widens proportionally to the jump height between phases. To resolve this problem, a novel variable interfacial parameter is introduced to the MM equation so that the interfacial transition layer is approximately constant regardless of any jumps between different phase levels. In this study, we propose an unconditionally stable numerical method for the modified MM equation with a variable interfacial parameter depending on the gradient of the phase-field function. The proposed numerical algorithm consists of the operator splitting scheme with a closed-form solution and the Fourier spectral method. To validate the performance of the proposed numerical method, we conduct a series of computational experiments. The numerical results confirm the superiority of the proposed numerical algorithm.

Keywords

Modica-Mortola (MM) functional / Interfacial function / Phase-field method / 65N06 / 65M06 / 39A14

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Funding

Korea University Grant

RIGHTS & PERMISSIONS

Shanghai University

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