In order to describe the nonlinear dynamics of a compliant bifurcated cardiovascular system, the reduced order 1D model of the arbitrary cardiovascular network is generated. Utilizing the physics-based nonlinear constitutive framework, this model can be adequately tested, calibrated and applied for patient-specific clinical diagnosis and prediction. The hyperbolic properties of the derived mathematical model are analyzed and used, constructing a monotone finite volume numerical scheme, with the second-order accuracy in time and space. The computational algorithm is able to simulate the fluid-structure interaction in pulsatile blood flow in the arterial system and is validated against well-known numerical, in-vitro and in vivo data from different cases of human arterial trees. The feasibility of tracking variations of cardiovascular markers, such as the blood pressure or stroke volume, have been previously studied experimentally by measurement of the pulse transit time. Here a new algorithm is also being presented for numerical calculation of the pulse transit time (PTT) for an essentially nonlinear wave that travels between two arbitrary sections of a cardiovascular tree. Presented method provides a more accurate estimate of the pulse transit time and it features a coupling algorithm to connect with commercial 3D CFD software (this feature will be added soon). These features make this work applicable for non-invasive estimation of central aortic pressure and stroke volume based on the peripheral measurements, and cuffless blood pressure predictions. The developed open - source software - entitled CardioFAN (Cardiovascular Flow Analysis) - is capable of multi-scale simulations, prediction of physiological parameters, calibration for a patient-specific cardiovascular tree, and diagnostics based on a high-resolution monotone finite volume numerical method. The code will accompany a manuscript with the same name, being submitted to the journal of "Biomechanics and Modeling in Mechanobiology". The user is encouraged to reference this code and the accompanying article (after potential publication).