  The theory of small-amplitude waves propagating across a blood vessel
junction has been well established with linear analysis. In this study we
consider the propagation of large-amplitude, nonlinear waves (i.e. shocks and
rarefactions) through a junction from a parent vessel into two (identical)
daughter vessels using a combination of three approaches: numerical
computations using a Godunov method with patching across the junction, analysis
of a nonlinear Riemann problem in the neighbourhood of the junction and an
analytical theory which extends the linear analysis to the following order in
amplitude. A unified picture emerges: an abrupt (prescribed) increase in
pressure at the inlet to the parent vessel generates a propagating shock wave
along the parent vessel which interacts with the junction. For modest driving,
this shock wave divides into propagating shock waves along the two daughter
vessels and reflects a rarefaction wave back towards the inlet. However, for
larger driving the reflected rarefaction wave becomes transcritical, generating
an additional shock wave. Just beyond criticality this new shock wave has zero
speed, pinned to the junction, but for further increases in driving this
additional shock divides into two new propagating shock waves in the daughter
vessels.
