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

• Applicants: Dr. Raul Borsche; Professor Dr. Stephan Trenn
• Subject Area: Mathematics
• Term: from 2016 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 314078707

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.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1962: Non-smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization
• International Connection: Netherlands