In this paper, we present a novel second order in time mixed finite element scheme for the Cahn-Hilliard-Navier-Stokes equations with matched densities. The scheme combines a standard second order Crank-Nicholson method for the Navier-Stokes equations and a modification to the Crank-Nicholson method for the Cahn-Hilliard equation. In particular, a second order Adams-Bashforth extrapolation and ...

In this paper, we consider nonlinear multidimensional Cahn–Hilliard and Kuramoto–Sivashinsky equations that have many important applications in physics chemistry, a certain natural generalization of these two to which refer as the generalized Cahn–Hilliard–Kuramoto–Sivashinsky equation. For an arbitrary number spatial independent variables, present complete list cases when latter equation admit...

Abstract: In a numerical study, we consider the Allen–Cahn equation with a double-obstacle constraint. The constraint is a multivalued function that is provided by the subdifferential of the indicator function on a closed interval. Therefore, performing a numerical simulation of our problem poses difficulties. We propose a new approximate method for the constraint and demonstrate its validity. ...

In this article, we study the energy dissipation property of time-fractional Allen-Cahn equation. We propose a decreasing upper bound that decreases with respect to time and coincides original at $t = 0$ as $t$ tends $\infty$. This can also be viewed nonlocal-in-time modified energy, summation an accumulation term due memory effect fractional derivative. particular, indicates indeed decays w.r....

In this note, we announce a general result resolving the long-standing question of nonlinear modulational stability, or stability with respect to localized perturbations, of periodic travelingwave solutions of the generalized Kuramoto–Sivashinski equation, establishing that spectral modulational stability, defined in the standard way, implies nonlinear modulational stability with sharp rates of...

We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension d ≤ 3, and study the semidiscretization in time of the equation by an implicit Euler method. We show that the method converges pathwise with a rate O(∆tγ) for any γ < 1 2 . We also prove that the scheme converges uniformly in the strong Lp-sense but wi...

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...