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# A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions

• * Corresponding author: Morgan Pierre

Dedicated to Michel Pierre on the occasion of his 70th birthday

• We propose a time semi-discrete scheme for the Caginalp phase-field system with singular potentials and dynamic boundary conditions. The scheme is based on a time splitting which decouples the equations and on a convex splitting of the energy associated to the problem. The scheme is unconditionally uniquely solvable and the energy is nonincreasing if the time step is small enough. The discrete solution is shown to converge to the energy solution of the problem as the time step tends to $0$. The proof involves a multivalued operator and a monotonicity argument. This approach allows us to compute numerically singular solutions to the problem.

Mathematics Subject Classification: Primary: 65M10, 35K67; Secondary: 65M60.

 Citation: • • Figure 1.  Solution $y_K$ for different values of $K$

Figure 2.  Solutions of the regularized problem without constraint ($y_K^ \varepsilon$, left) and with constraint ($\tilde{y}_K^ \varepsilon$, right)

Figure 3.  Initial condition $u_0$

Figure 4.  Solution $u(t)$ at time $t = 0.10$

Figure 5.  Solution $u(t)$ at singular time $t = 0.71$

Figure 6.  Stationary solution ($u(t)$ at time $t = 5.00$)

Figure 7.  Solution $y\mapsto u(t, x = 2, y)$ from $t = 0$ to $t = 5.00$

Table 1.  $L^2$-error and ratio of consecutive errors vs time step

 $m$ (cf. time step) 0 1 2 3 4 5 $L^2$-error 0.024 0.0124 0.0063 0.0032 0.0016 0.0008 ratio 1.94 1.97 1.97 2 2 —

Table 2.  Normalized CPU time vs time step for the linearly implicit (LI) scheme and the doubly splitting (DS) scheme

 $m$ (cf. time step) 0 1 2 3 4 5 LI scheme 1 2 4 8 16 32 DS scheme 165 262 305 381 511 489
• Figures(7)

Tables(2)

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