  In this paper, we propose a direct Eulerian generalized Riemann problem (GRP)
scheme for a blood flow model in arteries. It is an extension of the Eulerian
GRP scheme, which is developed by Ben-Artzi, et. al. in J. Comput. Phys.,
218(2006). By using the Riemann invariants, we diagonalize the blood flow
system into a weakly coupled system, which is used to resolve rarefaction wave.
We also use Rankine-Hugoniot condition to resolve the local GRP formulation. We
pay special attention to the acoustic case as well as the sonic case. The
extension to the two dimensional case is carefully obtained by using the
dimensional splitting technique. We test that the derived GRP scheme is second
order accuracy.
