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...

In this paper, a numerical solution of nonlinear partial differential equation, Benjamin-Bona-Mahony (BBM) and Cahn-Hilliard equation is presented by using Adomain Decomposition Method (ADM) and Variational Iteration Method (VIM). The results reveal that the two methods are very effective, simple and very close to the exact solution.

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...

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...

We prove a weak rate of convergence fully discrete scheme for stochastic Cahn--Hilliard equation with additive noise, where the spectral Galerkin method is used in space and backward Euler time. Compared Allen--Cahn type partial differential equation, error analysis here much more sophisticated due to presence unbounded operator front nonlinear term. To address such issues, novel direct approac...

For all n ≥ 1, we are interested in bounded solutions of the AllenCahn equation ∆u + u − u3 = 0 which are defined in all Rn+1 and whose zero set is asymptotic to a given minimal cone. In particular, in dimension n+1 ≥ 8, we prove the existence of stable solutions of the Allen-Cahn equation whose zero sets are not hyperplanes.