We consider a time semi-discretization of the Caginalp phase-field model based on an operator splitting method. For every time-step parameter τ , we build an exponential attractor Mτ of the discrete-in-time dynamical system. We prove that Mτ converges to an exponential attractor M 0 of the continuous-in-time dynamical system for the symmetric Hausdorff distance as τ tends to 0. We also provide an explicit estimate of this distance and we prove that the fractal dimension of Mτ is bounded by a constant independent of τ .

<abstract><p>We consider a time semidiscretization of the Ginzburg-Landau equation by the backward Euler scheme. For each time step $ \tau $, we build an exponential attractor of the dynamical system associated to the scheme. We prove that, as $ \tau $ tends to $ 0 $, this attractor converges for the symmetric Hausdorff distance to an exponential attractor of the dynamical system associated to the Allen-Cahn equation. We also prove that the fractal dimension of the exponential attractor and of the global attractor is bounded by a constant independent of $ \tau $.</p></abstract>

Our aim in this paper is to study on the Caginalp for a conserved phase-field with a polynomial potentiel of order 2p − 1. In this part, one treats the conservative version of the problem of generalized phase field. We consider a regular potential, more precisely a polynomial term of the order 2p − 1 with edge conditions of Dirichlet type. Existence and uniqueness are analyzed. More precisely, we precisely, we prove the existence and uniqueness of solutions.

This paper aims is to solve an advection problem where u=u(x, t) is the solution by Lax-Wendrof and finite difference methods, to study the analytical stability in L 2 [0, 1], L ∞ [0, 1], then calculate the truncation error of these methods and finally study the analytical convergence of these methods. These numerical techniques of resolution were implemented in Scilab.

This paper aims is to solve an advection problem where u=u(x, t) is the solution by Lax-Wendrof and finite difference methods, to study the analytical stability in L 2 [0, 1], L ∞ [0, 1], then calculate the truncation error of these methods and finally study the analytical convergence of these methods. These numerical techniques of resolution were implemented in Scilab.