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：本文介绍了为具有动态边界条件的半线性抛物线部分偏差方程式构建散状地块分割计划的新办法,拟议建造的基础是将该系统改制为局部差位数方程式,并为脱钩列入某些延迟条件。为了获得完全独立的计划,分拆办法与空间的有限元素和BDF的分离时间相结合。在本文件中,我们侧重于第二阶案,导致一个3美元的分步制。我们证明,在假设弱的CFL型条件的情况下,二级趋同,并通过数字实验证实了理论结论。此外,我们用数字实验来说明更高顺序分裂计划的可能性。