Coupling hyperbolic PDEs with switched DAEs: Analysis, numerics and application to blood flow models

双曲 PDE 与切换 DAE 的耦合：分析、数值及其在血流模型中的应用

• 批准号: 314078707
• 负责人: Dr. Raul Borsche
• 金额: --
• 依托单位: Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/314078707?language=en
• 关键词: Coupling; hyperbolic; PDEs; switched; DAEs

In this project we study hyperbolic partial differential equations (PDEs) with boundary conditions driven by switched differential algebraic equations (DAEs). This class of systems is motivated by models of the human circulatory system. The flow of blood in the vessels is described by a hyperbolic PDE, its connection to the heart is represented by boundary conditions. The dynamics of the heart can be modeled by a combination of ordinary differential equations and algebraic constraints. The corresponding choice depends on the state of the valves (e.g. when the valves are closed the flow is zero) which results in a switched DAE model.Due to the possible change of algebraic constraints at switching instants, solutions of switched DAEs exhibit jumps. Additionally, solutions may also contain Dirac impulses or their derivatives. The coupling of these discontinuities and Dirac-impulses with PDEs needs a rigorous solution theory and novel numerical schemes. Furthermore, the developed high order numerical methods will allow for more accurate simulations of the blood flow taking rigorously into account discontinuous and impulsive effects.

在这个项目中，我们研究了由切换微分代数方程驱动的带边界条件的双曲型偏微分方程。这类系统是由人体循环系统的模型驱动的。血液在血管中的流动由双曲线PDE描述，其与心脏的连接由边界条件表示。心脏的动力学可以通过常微分方程和代数约束的组合来建模。相应的选择取决于阀门的状态（例如，当阀门关闭时，流量为零），这导致切换DAE模型。由于切换时刻代数约束可能发生变化，切换DAE的解会出现跳跃。此外，解也可能包含狄拉克脉冲或其导数。这些不连续性和Dirac脉冲与偏微分方程的耦合需要一个严格的解决方案和新的数值方案。此外，开发的高阶数值方法将允许更准确地模拟血液流动，同时严格考虑不连续和脉冲效应。