Projection-difference implicit scheme for one-dimensional equations of hemodynamics

Source abstract

The paper is devoted to the construction and analysis of a projection-difference scheme for numerical simulation of blood flow in vascular systems. An implicit scheme is proposed for a hyperbolic system obtained by averaging the hydrodynamic equations over the vessel crosssection. The unconditional stability of the proposed scheme is demonstrated. For practical implementation, a method of splitting with respect to physical processes is proposed. The first stage of the method involves computation of the deformed configuration of the vessel for a given flow velocity. The second stage involves computation of the fluid velocity in the deformed vessel. Riemann invariants are used to obtain the values at the boundaries. The FreeFEM++ package is used for the software development. As a result of computations for the scheme with piecewise-linear nodal functions, it is demonstrated that the convergence with second order takes place. As a special case, a scheme with weights is considered. It is demonstrated, that the variations of the weight parameter lead to monotonization of the numerical solutions.