This paper introduces a novel approach for the construction of bulk--surface splitting schemes for semi-linear parabolic partial differential equations with dynamic boundary conditions. The proposed construction is based on a reformulation of the system as a partial differential--algebraic equation and the inclusion of certain delay terms for the decoupling. To obtain a fully discrete scheme, the splitting approach is combined with finite elements in space and a BDF discretization in time. Within this paper, we focus on the second-order case, resulting in a $3$-step scheme. We prove second-order convergence under the assumption of a weak CFL-type condition and confirm the theoretical findings by numerical experiments. Moreover, we illustrate the potential for higher-order splitting schemes numerically.

翻译：本文提出了一种新颖的方法，用于构造半线性抛物型偏微分方程（含动态边界条件）的体-面分离格式。该构造方法基于将系统重新表述为偏微分-代数方程，并引入特定延迟项以实现解耦。为获得全离散格式，我们结合了空间有限元方法与时间方向上的BDF离散化。本文重点研究二阶情形，由此产生了一个三步格式。在弱CFL型条件下，我们证明了二阶收敛性，并通过数值实验验证了理论结果。此外，我们通过数值算例展示了高阶分离格式的潜力。