  The purpose of this work is to construct a simple, efficient and accurate
well-balanced numerical scheme for one-dimensional (1D) blood flow in large
arteries with varying geometrical and mechanical properties. As the steady
states at rest are not relevant for blood flow, we construct two well-balanced
hydrostatic reconstruction techniques designed to preserve low-Shapiro number
steady states that may occur in large network simulations. The Shapiro number S
h = u/c is the equivalent of the Froude number for shallow water equations and
the Mach number for compressible Euler equations. The first is the low-Shapiro
hydrostatic reconstruction (HR-LS), which is a simple and efficient method,
inspired from the hydrostatic reconstruction technique (HR). The second is the
subsonic hydrostatic reconstruction (HR-S), adapted here to blood flow and
designed to exactly preserve all subcritical steady states. We systematically
compare HR, HR-LS and HR-S in a series of single artery and arterial network
numerical tests designed to evaluate their well-balanced and wave-capturing
properties. The results indicate that HR is not adapted to compute blood flow
in large arteries as it is unable to capture wave reflections and transmissions
when large variations of the arteries' geometrical and mechanical properties
are considered. On the contrary, HR-S is exactly well-balanced and is the most
accurate hydrostatic reconstruction technique. However, HR-LS is able to
compute low-Shapiro number steady states as well as wave reflections and
transmissions with satisfying accuracy and is simpler and computationally less
expensive than HR-S. We therefore recommend using HR-LS for 1D blood flow
simulations in large arterial network simulations.
