A Semi-Implicit Finite Volume Scheme for Blood Flow in Elastic and Viscoelastic Vessels

A novel staggered semi-implicit finite volume method for the simulation of one-dimensional blood flow in networks of elastic and viscoelastic vessels is proposed. The one-dimensional blood flow model is split into three subsystems: one containing the nonlinear convective terms, a second one for the viscoelastic terms and a final subsystem for the pressure-related terms. An explicit approach is then employed to discretize the convective subsystem, while the viscous and pressure terms are treated implicitly. This leads to a CFL-type time step restriction which depends only on the bulk velocity of the flow and not on the speed of the pressure waves or on the visco elasticity. As such, the method is computationally efficient in case of low speed index, which is equivalent to a low Mach regime for the Navier-Stokes equations. To extend the proposed methodology to the case of networks, a novel and very simple 3D approach for the treatment of junctions is proposed. Each junction is represented by a unique three-dimensional primal cell and the Euler equations are employed to approximate the velocity and pressure unknowns. A multidimensional numerical flux then takes into account the elementary information of the junction geometry, namely normal vectors and areas of the incident vessels, when approximating the blood flow at the cell interfaces embedded in the 3D cell. As such, the final scheme, based on guaranteeing mass and momentum conservation through the junction, is able to capture reflecting waves properly considering the effect of the different incident angles of vessels at a junction. The proposed methodology is carefully validated in the context of single elastic and viscoelastic vessels and small networks showing a good agreement with available analytical, experimental and numerical data.