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

在这个项目中，我们研究了边界条件由切换微分代数方程驱动的双曲型偏微分方程。这类系统是由人体循环系统模型驱动的。血管中的血流用双曲PDE来描述，它与心脏的连接用边界条件来表示。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），这会导致DAE模型的切换。由于切换时刻的代数约束可能发生变化，切换DAEs的解呈现跳变。此外，解也可能包含狄拉克脉冲或其导数。这些不连续面和狄拉克脉冲与偏微分方程的耦合需要严格的求解理论和新颖的数值格式。此外，开发的高阶数值方法将允许更准确地模拟血流，严格考虑不连续和脉冲效应。

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.