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.

中文翻译：

---

具有奇异势和动态边界条件的Caginalp系统的双分裂方案

我们提出了具有奇异电位和动态边界条件的Caginalp相场系统的时间半离散方案。该方案基于将方程解耦的时间分割以及与问题相关的能量的凸分割。如果时间步长足够小，则该方案可以无条件地唯一解决，并且能量也不会增加。随着时间步长趋于$ 0 $，离散解决方案收敛到问题的能量解。证明涉及一个多值运算符和一个单调性参数。这种方法使我们能够计算出该问题的数字奇异解。