Decoupling integration schemes of higher order for poroelastic networks

多孔弹性网络的高阶解耦积分方案

• 批准号: 467107679
• 负责人: Professor Dr. Robert Altmann
• 金额: --
• 依托单位: Institut für Analysis und Numerik (IAN)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/467107679?language=en
• 关键词: Decoupling; integration; schemes; higher; order

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.

多孔弹性多网络模型广泛应用于地学、生物医学等领域。例如，对脑水肿的研究，考虑了不同的血液循环和脑脊液，总共产生了四种液体比较。相应的方程组是椭圆型和抛物型的耦合偏微分方程组，特别是在三维情况下，如果使用标准方法，在计算上是困难的，甚至是不可行的。该项目旨在构建一类新的高效的高阶积分方案，该方案结合了单片方法的简单性和迭代方法的巨大的解耦速度。由于耦合偏微分方程组的空间离散化导致了一个微分-代数方程(DAE)，因此收敛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提出的想法推广到非线性情况。收敛分析基于这样一个事实，即半显式格式可以解释为相关延迟方程的隐式格式，其中时间延迟等于步长。其次，通过寻找合适的延迟方程来构造高阶半显式格式，使得延迟方程与原始模型之间的误差达到期望的阶数。实现和分析了一种自适应步长选择机制，进一步提高了方法的效率。由于收敛分析依赖于所谓的弱耦合条件，第三，我们证明了耦合条件与相关时滞方程的渐近稳定性直接相关。利用这种联系，我们将建立半显式时间步长方法的收敛与中立型延迟方程和延迟DAE的渐近稳定性之间的等价结果，从而将不同的研究群体联系起来。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。该项目是两个申请者在耦合偏微分方程的时间离散化方案和时延系统分析方面的互补专业知识的协同合作。