The Allen-Cahn equation is a gradient system, where the free-energy func4 tional decreases monotonically in time. We develop an energy stable reduced order 5 model (ROM) for a gradient system, which inherits the energy decreasing property 6 of the full order model (FOM). For the space discretization we apply a discontinuous 7 Galerkin (dG) method and for time discretization the energy stable av...

We consider two-phase flow problems, modelled by the Cahn-Hilliard equation. In our work, the nonlinear fourth-order equation is decomposed into a system of two second-order equations for the concentration and the chemical potential. We analyse solution methods based on an approximate two-by-two block factorization of the Jacobian of the nonlinear discrete problem. We propose a preconditioning ...

■ Abstract The phase-field method has recently emerged as a powerful computational approach to modeling and predicting mesoscale morphological and microstructure evolution in materials. It describes a microstructure using a set of conserved and nonconserved field variables that are continuous across the interfacial regions. The temporal and spatial evolution of the field variables is governed b...

For the Allen-Cahn equation, it is well known that there a monotone standing wave joining with balanced wells of potential. In this paper we study existence traveling solutions for equation on an infinite channel. Such possess large number oscillation and they are obtained aid variational arguments.

We define a (mean curvature flow) entropy for Radon measures in $\mathbb{R}^n$ or compact manifold. Moreover, we prove monotonicity formula of the associated with parabolic Allen-Cahn equations. If ambient manifold is non-negative sectional and parallel Ricci curvature, this consequence new equation. As an application, show that when initial data small enough (less than twice energy one-dimensi...