Flow of Blood: Coupling hyperbolic PDEs with switched DAEs

Modeling the flow of blood in the human circulatory system is a challenging task involving many mathematical aspects.

In this joined project of the TU Kaiserslautern and the University of Groningen funded by the DFG SPP1962 Priority Program  we study hyperbolic partial differential equations (PDEs) with boundary conditions driven by switched differential algebraic equations (DAEs).

The flow of blood in the vessels is described by hyperbolic PDEs, 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 appropriate 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.

If you are interested in further information please contact Raul Borsche (borsche@mathematik.uni-kl.de).