In this paper, we present an efficient numerical method for multiphase image segmentation using a multiphase-field model. The method combines the vector-valued Allen– Cahn phase-field equation with initial data fitting terms containing prescribed interface width and fidelity constants. An efficient numerical solution is achieved using the recently developed hybrid operator splitting method for ...

The bidomain model is the standard model describing electrical activity of the heart. Here we study the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain AllenCahn equation) in two spatial dimensions. In the bidomain Allen-Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the ...

We study the asymptotic dynamics of the Cahn–Hilliard equation via the “Gamma-convergence” of gradient flows scheme initiated by Sandier and Serfaty. This gives rise to an H1-version of a conjecture by De Giorgi, namely, the slope of the Allen–Cahn functional with respect to the H−1-structure Gamma-converges to a homogeneous Sobolev norm of the scalar mean curvature of the limiting interface. W...

We study two novel decoupled energy-law preserving numerical schemes for solving the CahnHilliard-Darcy (CHD) system which models two-phase flow in porous medium or in a Hele-Shaw cell. In the first scheme, the velocity in the Cahn-Hilliard equation is treated explicitly so that the Darcy equation is completely decoupled from the Cahn-Hilliard equation. In the second scheme, an intermediate vel...

In this paper, we develop a high order semi-implicit time discretization method for highly nonlinear PDEs, which consist of the surface diffusion and Willmore flow of graphs, the Cahn-Hilliard equation and the Allen-Cahn/Cahn-Hilliard system. These PDEs are high order in spatial derivatives, which motivates us to develop implicit or semi-implicit time marching methods to relax the severe time s...

Spatially localized, time-periodic structures are common in pattern-forming systems, appearing in fluid mechanics, chemical reactions, and granular media. We examine the existence of oscillatory localized states in a PDE model with single frequency time dependent forcing, introduced in [22] as phenomenological model of the Faraday wave experiment. In this study, we reduce the PDE model to the f...

The phase field method is playing an increasingly important role in understanding and predicting morphological evolution materials biological systems. Here, we develop a new analytical approach based on the bifurcation analysis to explore mathematical solution structure of models. Revealing such structures not only great interest but also may provide guidance experimentally or computationally u...

In this paper, we consider the vector-valued Allen–Cahn equations which model phase separation in N-component systems. The considerations of solving numerically the vector-valued Allen–Cahn equations are as follows: (1) the use of a small time step is appropriate to obtain a stable solution and (2) a sufficient number of phase-field variables is required to capture the correct dynamics. However...