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$\varepsilon\frac{(U^{n+1}-2U^n+U^{n-1}}{\Delta t^2} +\frac{U^{n+1}-U^n}{\Delta t}+\nabla F(U^{n+1})=G^{n+1}, n\ge 0, $

where $\Delta t>0$ is the time step, $\varepsilon\ge 0$, $(G^{n+1})_n$
is a sequence in $ R^d$ which converges to $0$ in a suitable way,
and $F\in C^{1,1}_{l o c}(R^d, R)$ is a function which satisfies a Łojasiewicz inequality.
We prove that the above scheme is Lyapunov stable and that any bounded sequence $(U^n)_n$
which complies with it converges to a critical point of $F$ as $n$ tends to $\infty$.
We also obtain convergence rates. We assume that $F$ is semiconvex for some constant $c_F\ge 0$
and that $1/\Delta t

We consider a time semi-discretization of the Caginalp phase-field model based on an operator splitting method. For every time-step parameter $ \tau $, we build an exponential attractor $ \mathcal{M}_\tau $ of the discrete-in-time dynamical system. We prove that $ \mathcal{M}_\tau $ converges to an exponential attractor $\mathcal{M}_0$ of the continuous-in-time dynamical system for the symmetric Hausdorff distance as $ \tau $ tends to $0$. We also provide an explicit estimate of this distance and we prove that the fractal dimension of $ \mathcal{M}_\tau $ is bounded by a constant independent of $ \tau $.

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