An unconditionally stable splitting method for the Allen-Cahn equation with logarithmic free energy

Abstract

We present an unconditionally stable splitting method for the Allen-Cahn (AC) equation with logarithmic free energy which is more physically meaningful than the commonly used polynomial potentials. However, owing to the singularity of the logarithmic free energy, it is difficult to develop unconditionally stable computational methods for the AC equation with logarithmic potential. To overcome this difficulty, prior works added a stabilizing term to the logarithmic energy or used a regularized potential. In this study, the AC equation with logarithmic potential is solved by using an operator splitting method without adding a stabilizing term nor regularizing the logarithmic energy. The equation involving logarithmic free energy potential is solved using an interpolation method; the other diffusion equation is solved numerically by applying a finite difference method. Each solution algorithm is unconditionally stable, the proposed scheme is unconditionally stable. Various computational experiments demonstrate the performance of the proposed method.