An entropy stable and well-balanced scheme for an augmented blood flow model with variable geometrical and mechanical properties

The flow of blood through a vessel can be described by a hyperbolic system of balance equations for the cross-sectional area and averaged velocity as functions of axial spatial position and time. The variable arterial wall rigidity and the equilibrium cross-sectional area are incorporated within the so-called tube law that gives rise to an internal pressure term. This system can be written as a conservative hyperbolic system for five unknowns. An entropy stable scheme for this augmented one-dimensional blood flow model is developed based on entropy conservative numerical flux. It is proved that the proposed scheme is well-balanced in the sense that it preserves both trivial (zero velocity) and non-trivial (non-zero velocity) steady-state solutions. Several demanding numerical tests show that the scheme can handle various kinds of shocks and preserves stationary solutions when geometrical and mechanical properties of the vessel are variable.