This paper introduces a novel approach for the construction of bulk–surface splitting schemes for semilinear 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 backward differentiation formula 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.

中文翻译：

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具有动态边界条件的抛物线问题的二阶体表面分裂

本文介绍了一种为具有动态边界条件的半线性抛物型偏微分方程构造体表面分裂格式的新方法。所提出的结构基于将系统重新表述为偏微分代数方程，并包含某些解耦延迟项。为了获得完全离散的方案，分裂方法与空间上的有限元和时间上的后向微分公式相结合。在本文中，我们重点关注二阶情况，从而得出 $3$ 步方案。我们证明了弱 CFL 型条件假设下的二阶收敛性，并通过数值实验证实了理论结果。此外，我们还以数值方式说明了高阶分裂方案的潜力。