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

在这个项目中，我们研究了由切换微分代数方程(DAE)驱动的带边界条件的双曲型偏微分方程(PDE)。这类系统是由人体循环系统的模型驱动的。血管中的血液流动用双曲线偏微分方程来描述，它与心脏的联系用边界条件来表示。心脏的动力学可以用常微分方程式和代数约束相结合的方法来建模。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)，这导致了切换DAE模型。由于切换瞬间代数约束的可能变化，切换DAE的解表现出跳跃。此外，解还可能包含狄拉克脉冲或其导数。这些间断和狄拉克脉冲与偏微分方程组的耦合需要严格的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液流动，并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.