A great deal of qualitative and quantitative research has be done on the reactiondiffusion problems (see [16], [17] and references therein), in particular on the AllenCahn equation (see [1], [10], [11], [15]) and the phase-field transition system (see [4], [5], [6], [13]). For numerical investigations on the phase-field system and nonlinear Allen-Cahn equation, we refer to the works [2], [8], [12], [13], [14] and [15].

Up to our knowledge, the classical Allen-Cahn equation supplied with mixed boundary conditions has not been considered numerically. Thus our first aim is to fill this gap. Also we have to observe that a Lyapunov energy functional for this case is no longer available. We also generalize this model using high order polynomials as nonlinear terms. These polynomials conserve the dynamic of the classical AllenCahn model, i.e., $u=0$ is an unstable state and $u=\pm 1$ are stable states. The periodic case has been recently addressed in [10].

In the second part we extend our analysis to a more involved model, namely the so called Caginalp model. This model add to the previous nonlinearity a genuine
stiffness. In order to resolve this stiffness we make use of a multistep method of the numerical differentiation formula type. Our experience with spectral collocation gathered in the recent booklet [7] encouraged us to use this discretization method based on Chebyshev polynomials.

Let’s consider the nonlinear initial-boundary value parabolic problem in $Q:=[-1,1]\times(0,T]$ with $T>0$

where $\tau,\xi$ and $c_{0}$ are positive physical parameters. The problem is the 1D variant of the general formulation provided in [15]. Existence, uniqueness and regularity results are also available in this paper. Corresponding to simple Dirichlet problem, a recent paper is that of Miranville [11].

For the nonlinear term $F$, we have chosen the polynomial form (Allen-Cahn)

and the source term $g$ was considered as null in the first instance. The case of the greatest interest is that when the measure of interface thickness $\xi$ is small, i.e. $0<\xi\ll 1$.

A physical (correct) range of variation for relaxation time $\tau$ and for the measure of interface thickness $\xi$ is suggested by the inequality (7) from Section 3.

The physics of the solidification process (see [4], [8], [13]), leads to the following system of genuinely nonlinear parabolic differential equations

where

and the source term is

The unknown $\varphi$ is the so called phase field function and $u$ is the difference between the temperature $T$ and its melting value $T_{M}$ i.e., $u(x,t):=T(x,t)-T_{M}(x,t)$. The
physical parameters $\ell,\xi$, and $\tau$ stand respectively for latent heat per unit mass, length scale and relaxation time. $K$ is the thermal diffusivity (see also [8], [13]).

In [4] the author has carried out a rigorous analysis concerning the existence and regularity of solutions of (6). Invariant regions of the solution space have also been obtained. He observes that the existence has been shown for fixed values of $\xi$ and $\tau$ satisfying the stability inequality

He also remarks that one of the questions of interest is the behavior of these equations in the limit of small $\xi$ and $\tau$. Boundary as well as initial conditions have to be attached to (6). Actually, we have solved the initial-Dirichlet boundary value problem

In some of our numerical experiments we have used, as in Section $2.1\xi=1/15$, $\tau=1/10$ and then $K=1$ and $\ell=1$. However, in order to observe the behavior of the solutions of (6) for $\xi^{2}/\tau\rightarrow 0$ we also have considered this problem when $\xi^{2}/\tau=10^{-3}$ ( $\xi^{2}=10^{-2},\tau=10$ ). An analogous semi-discretized problem with that attached to Allen-Cahn equation, i.e., (3), but double as dimension, is used to solve (6).

The interval of time integration has been $T=10$. In this case the problem is a fairly stiff one and consequently the Runge-Kutta integrator failed. The results presented in Fig. 8 and Fig. 9 have been carried out using the MATLAB code ode15s which is a multi-step solver based on a numerical differentiation formula.

However, this scheme for time marching along with ChC provided accurate numerical results. The main remark refers to the fact that in this case the transition to meta-stability is not so smooth as it has proved to be in Allen-Cahn case (compare Fig. 7 and Fig. 8).

The evolution of temperature variable is fairly predictable. After a short transition period, the initial sinusoidal perturbation becomes a linear variation between the cold or solid state and the hot or liquid one. In fact the temperature variable appears linearly in the system (6).

The Chebyshev collocation along with Runge-Kutta scheme or a numerical differentiation scheme for the stiff Caginalp’s model provided accurate solutions to the problems considered in this study. In both situations the algorithm is stable, i.e., provides identical results for values of the cut off parameter $N$ in the range $2^{5}$ and $2^{8}$.

The metastability phenomenon has been accurately captured in both Allen-Cahn and Caginalp’s models. Neither the higher nonlinearities nor the mixed boundary conditions do not have modified significantly the behavior of solutions in the former model.

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