Decoupling integration schemes of higher order for poroelastic networks

多孔弹性多网络模型出现在各种不同的应用领域，包括地球科学和生物医学。例如，对脑水肿的调查说明了不同的血液周期和脑脊液，从而得出总共四个液体室。相应的系统是椭圆型和抛物型的耦合偏微分方程（PDE），特别是在三维情况下，如果采用标准方法，在计算上是具有挑战性的，甚至是不可行的。该项目旨在构建一类新的高阶高效积分方案，将单片方法的简单性与解耦问题的迭代方法的巨大加速相结合。由于耦合偏微分方程的空间离散化结果是一个微分代数方程（DAE），因此收敛性分析是一项具有挑战性的任务。该项目的第一个主要贡献是将申请人最近介绍的思想扩展到非线性情况。收敛性分析是基于半显式格式可以解释为相关延迟方程的隐式格式，其中时滞等于步长。其次，通过寻找合适的延迟方程，构造高阶半显式格式，使延迟方程与原模型之间的误差符合期望阶；自适应步长选择机制的实现和分析进一步提高了方法的效率。由于收敛分析依赖于一个所谓的弱耦合条件，我们第三，通过证明它与相关延迟方程的渐近稳定性直接相关来证明耦合条件。利用这种联系，我们将建立半显式时间步进方法的收敛性与中立型延迟方程和延迟DAEs的渐近稳定性之间的等价结果，从而连接不同的研究群体。第四，在数据正则性的弱假设下，分析了相关时滞方程解的正则性。该项目是两位申请人在耦合偏微分方程的时间离散化方案和时滞系统分析方面的互补专业知识的协同合作。

Poroelastic multiple-network models arise in a variety of different application domains, including geoscience and biomedicine. The investigation of cerebral edema, for instance, accounts for different blood cycles and a cerebrospinal fluid, resulting in a total of four fluid comparments. The corresponding system is a coupled partial differential equation (PDE) of elliptic and parabolic type that, in particular in the 3-dimensional case, is computationally challenging or even unfeasible if standard methods are applied. This project aims to construct a novel class of highly efficient integration schemes of higher order that combine the simplicity of monolithic approaches with the tremendous speed-ups of iterative methods that decouple the problem. Since the spatial discretization of the coupled PDE results in a differential-algebraic equation (DAE), the convergence analysis renders a challenging task. The first main contribution of the project is the extension of recently introduced ideas of the applicants to the nonlinear case. The convergence analysis is based on the fact that the semi-explicit scheme can be interpreted as an implicit scheme for a related delay equation, where the time-delay equals the step size. Second, higher-order semi-explicit schemes are constructed by finding suitable delay equations, such that the error between the delay equation and the original model is of the desired order. Implementation and analysis of an adaptive step size selection mechanism further improve the method's efficiency. Since the convergence analysis relies on a so-called weak coupling condition, we, third, justify the coupling condition by showing that it is directly related to the asymptotic stability of the related delay equation. Exploiting this connection, we will establish an equivalence result between the convergence of semi-explicit time-stepping methods and the asymptotic stability of neutral delay equations and delay DAEs, thus, connecting the different research communities. Fourth, we analyze the regularity of solutions of the related delay equations under weak assumptions on the regularity of the data. The project is a synergistic collaboration of the two applicants' complementary expertise on time discretization schemes for coupled PDEs and analysis of time-delay systems.