Long-time stability of the implicit Euler scheme for a three dimensional globally modified two-phase flow model

Abstract

In this article we study a globally modified Allen–Cahn–Navier–Stokes system in a three-dimensional domain. The model consists of the globally modified Navier–Stokes equations proposed in ( Adv. Nonlinear Stud. 6 ( 2006 ) 411–436) for the velocity, coupled with an Allen–Cahn model for the order (phase) parameter. We discretize these equations in time using the implicit Euler scheme and we prove that the approximate solution is uniformly bounded. We also show that the sequence of the approximate solutions of the globally modified Allen–Cahn–Navier–Stokes system converges, as the parameter N goes to infinity, to the solution of the corresponding discrete two-phase flow system. Using the uniform stability of the scheme and the theory of the multi-valued attractors, we then prove that the discrete attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.

Approximation of the Long-Time Dynamics of the Dynamical System Generated by the Ginzburg-Landau Equation

“…As in [2] (see also [3,5]), one can prove the following Theorem 5.3. For any k > 0, the multivalued map S k associated to the implicit Euler scheme (2.5) generates a closed discrete multivalued semigroup {S k } n∈N .…”

“…In this section, we first recall some results on the theory of the multi-valued attractors (see [2,3,5]) and then we apply them to our model.…”